FOUNDATION COURSE FOR CSE

MATHEMATICS OPTIONAL

ANALYTIC GEOMETRY
 PREVIOUS YEAR QUESTIONS (2008-2022)
 ASSIGNMENTS

By
Avinash Singh (Ex-IES)
B.Tech, IIT Roorkee

DELHI CENTRE BHOPAL CENTRE

27-B, Pusa Road, Metro Pillar No. 118, Plot No. 46 Zone - 2, M.P Nagar,
Near Karol Bagh Metro, New Delhi-110060 Bhopal - 462011

Phone: 8081300200, 8827664612 | E-mail : info@nextias.com | Web: www.nextias.com

2 Mathematics (Optional) Foundation Course for CSE

PREVIOUS YEAR QUESTIONS

ANALYTIC GEOMETRY
PLANE
1. The plane x – 2y + 3z = 0 is rotated through a right angle about its line of intersection with the plane 2x + 3y – 4z – 5 = 0. Find
the equation of the plane in its new position.

2. Find the equation of the plane which passes through the points (0, 1, 1) and (2, 0, –1) and is parallel to the line joining
the points (–1, 1, –2), (3, –2, 4). Find also the distance between the line and the plane.

3. Obtain the equation of the plane passing through the point (2, 3, 1) and (4, 5, 3) parallel to x-axis.

4. Find the equation of the plane parallel to 3x – y + 3z = 8 and passing through the point (1, 1, 1).

STRAIGHT LINE

x2 y2
1. A line is drawn through a variable point on the ellipse,   1 , z = 0 to meet two fixed lines y = mx, z = c and
a2 b2
y = –mx, z = –c. Find the locus of the line.

2. Find the equations of the straight line through the point (3, 1, 2) to intersect the straight line x + 4 = y + 1 = 2(z + 2) and
parallel to the plane 4x + y + 5z = 0.

3. Prove that two of the straight lines represented by the equation x3 + bx2y + cxy2 + y3 = 0 will be right angles if
b + c = –2.

x1 y 2
4. Find the shortest distance between the line   z  3 and y – mx = z = 0. For what values of 'm', will the two
2 4
lines intersect.

5. Find the surface generated by a line which intersects the lines y = a = z, x + 3z = a = y + z and parallel to the plane
x + y = 0.

6. Find the shortest distance between the skew lines:

x3 8y z3 x3 y7 z6

  and  
3 1 1 3 2 4

7. Find the shortest distance between the lines a1x + b1y + c1z + d1 = 0, a2x + b2y + c2z + d2 = 0 and the z-axis.

x1 y1 z1

8. Find the projection of straight line   on the plane x + y + 2z = 6.
2 3 1

x1 y3 z2 x y 7 z7

9. Show that the lines intersect   and   . Find the intersection point and the equation
3 2 1 1 3 2

of the plane containing them.

 Analytic Geometry 3

SPHERE
1. A sphere S has points (0, 1, 0), (3, –5, 2) at opposite ends of a diameter. Find the equation of the sphere having the
intersection of the sphere S with the plane 5x – 2y + 4z + 7 = 0 as great circle. (20)

2. Find the equations (in symmetric form) of the tangent line to the circle x2 + y2 + z2 + 5x – 7y + 2z – 8 = 0, 3x –2y + 4z +
3 = 0 at the point (–3, 5, 4). (12)

3. Find the equation of the sphere having its centre on the plane 4x – 5y –z = 3 and passing through the circle
x2 + y2 + z2 – 12z – 3y + 4z + 8 = 0, 3x + 4y – 5z + 3 = 0. (12)

4. Show that every sphere through the circle x2 + y2 – 2ax + r2 = 0, z = 0 cuts orthogonally every sphere through the circle
x2 + z2 = r2, y = 0 (20)

5. Show that the plane x + y – 2z = 3 cuts the sphere x2 + y2 + z2 – x + y = 2 in a circle of radius 1 and find the equation of
the sphere which has this circle as a great circle.

6. Show that the equation of the sphere which touches the sphere 4(x2 + y2 + z2) + 10x – 25y – 2z = 0 at the point
(1, 2, –2) and passes through the point (–1, 0, 0) is x2 + y2 + z2 + 2x – 6y + 1 = 0. (10)

7. A sphere S has points (0, 1, 0), (3, –5, 2) at opposite ends of a diameter. Find the equation of the sphere having the
intersection of the sphere S with the plane. 5x – 2y + 4z + 7 = 0 as a great circle.

8. Find the coordinates of the points on the sphere x2 + y2 + z2 – 4x + 2y = 4, the tangent planes at which are parallel to the
plane 2x – y + 2z = 1. (10)

9. For what positive value of 'a', the plane ax – 2y + z + 12 = 0 touches the sphere x2 + y2 + z2 – 2x – 4y + 2z – 3 = 0 and hence
find the point of contact. (10)

10. Find the equation of the sphere which passes through the circle x2 + y2 = 4 ; z = 0 and is cut by the plane. x + 2y + 2z =
0 in a circle of radius 3. (10)

11. A plane passes through a fixed point (a, b, c) and cuts the axes at the points A, B, C respectively. Find the locus of the
centre of the sphere which passes through the origin O and A, B, C. (15)

12. Find the equation of the sphere in xyz plane passing through the points (0, 0, 0), (0, 1, –1), (–1, 2, 0) and (1, 2, 3). (12)

13. (i) The plane x + 2y + 3z = 12 cuts the axes of coordinates in A, B, C. Find the equations of the circle circumscribing the
triangle ABC.

(ii) Prove that the plane z = 0 cuts the enveloping conc. of the sphere x2 + y2 + z2 = 11 which has the vertex at
(2, 4, 1) is a rectangular hyperbola.

CONE + CYLINDER

x 2 y 2 z2
1. Find the length of the normal chord through a point P of the ellipsoid   = 1 and prove that if it is equal to
a2 b2 c 2
4PG3 where G3 is the point where the normal chord through P meets the xy plane, then P lies on the cone:

x y2 z2
6
(2c 2  a 2 )  6 (2c 2  b 2 )  4 = 0. (15)
a b c

2. Find the equation of the cone with (0, 0, 1) as the vertex and 2x2 – y2 = 4, z = 0 as the guiding curve. (13)

3. Show that the cone 3yz – 2zy – 2xy = 0 has an infinite set of three mutually perpendicular generators.
4 Mathematics (Optional) Foundation Course for CSE

4. If 6x = 3y = 2z represents one of the three mutually perpendicular generators of the cone 5yz – 8zx – 3xy = 0 then obtain
the equations of the other two generators. (13)

5. Examine whether the plane x + y + z = 0 cuts the cone yz + zx + xy = 0 in perpendicular lines. (13)

u2 v 2 w 2
6. Prove that the equation, ax2 + by2 + cz2 + 2ux + 2vy + 2wz + d = 0, represents a cone if   = d. (13)
a b c

7. A cone has for its guiding curve the circle x2 + y2 + 2ax + 2by = 0, z = 0 and passes through a fixed point (0, 0, c). If the
section of the cone by the plane y = 0 is a rectangular hyperbola, prove that the vertex lies on the fixed circle x2 + y2 + z2
+ 2ax + 2by = 0, 2ax + 2by + cz = 0. (15)

x y z
8. A variable plane is parallel to the plane   = 0 and meets the axes in A, B, C respectively. Prove that the circle
a b c

b c c a a b
ABC lies on the cone yz     zx     xy    = 0 (20)
c b a c b a

9. Show that the cone yz + zx + xy = 0 cuts the sphere x2 + y2 + z2 = a2 in two equal circles, and find their area.

x y z
10. If   represent one of a set of three mutually perpendicular generators of the cone 5yz – 8zx – 3xy = 0, find the
1 2 3

equations of the other two. (20)

CONICOID + PARABOLOID
1. Prove that in general, three normals can be drawn from a given point to the paraboloid x2 + y2 = 2az, but if the point
lies on the surface 27a (x2 + y2) + 8(a –z)2 = 0 then two of the three normals coincide. [15]

2. Find the equation of the tangent plane at point (1, 1, 1) to the conicoid 3x2 – y2 = 2z. [10]

3. Find the locus of the point of intersection of three mutually perpendicular tangent planes to ax2 + by2 + cz2 = 1.

[10]

4. Two perpendicular tangent planes to the paraboloid x2 + y2 = 2z intersect in a straight line in the plane x = 0.
Obtain the curve to which this straight line touches. [15]

5. Show that the lines drawn from the origin parallel to the normals to the central conicoid ax2 + by2 + cz2 = 1, at its

2
 x 2 y 2 z 2   lx my nz 
point of intersection with the plane lx + my + nz = p generate the cone p       
2
[15]
 a b c   a b c 

6. Show that the locus of a point from which the three mutually tangent lines can be drawn to the paraboloid
x2 + y2 + 2z = 0 is x2 + y2 + 4z = 1. [20]

x 2 y 2 z2
. Three points P, Q, R are taken on the elliploid    1 so that the lines joining P, Q, R to the origin are
a2 b 2 c 2

mutually perpendicular. Prove that the plane PQR touches a fixed sphere. [20]

5
8. Show that the plane 3x  4 y  7 z   0 touches the paraboloid 3x 2  4 y 2  10 z and find the point of contact.
2
 Analytic Geometry 5
x2 y 2
9. Prove that the normals from the points (   ) to the paraboloid   2 z lies on the cone
a2 b2

  a2  b2
  0.
x  y  z

10. Show that the enveloping cylinders of the ellipsoid ax 2  by 2  cz 2  1 with generators perpendicular to z-axis
meet the plane z = 0 in parabolas.

GENERATING LINES

1. Find the equations to the generating lines of the paraboloid  x  y  z  2 x  y  z   6 z which pass through the
point (1, 1, 1). [13]

2. Find the equations of the two generating lines through any point (a cos , b sin , 0) of the principal elliptic section
x2 y2
  1, z  0, of the hyperboloid by the plane z = 0.
a2 b 2

3. A variable generator meets two generators of the system through the extremities B and B of the minor axis of the

x 2 y 2 z2
principal elliptic section of the hyperboloid 2  2  2  1 in P and P. Prove that BP . B P  a 2  c 2 . [20]
a b c

4. Show that the generators through any one of the ends of an equi-conjugate diameter of the principal elliptic
x 2 y 2 z2
section of the hyperboloid    1 are inclined to each other at an angle of 60° if a2 + b2 = 6c2. Find also
a2 b 2 c 2
the condition for the generators to be perpendicular to each other. [20]

REDUCTION OF GENERAL 2nd DEGREE EQUATION

1. Reduce the following equation to the standard form and hence determine the nature of the conicoid:

x 2  y 2  z 2  yz  zx  xy  3x  6 y  9 z  21  0
6 Mathematics (Optional) Foundation Course for CSE

ASSIGNMENT
ANALYTIC GEOMETRY
BASICS
1. Find the locus of a point which moves so that sum of its distances from the points (a, 0, 0) and (–a, 0, 0) is constant.

 a2  2
Ans. x 2  1  2  y  z 2  k 2  a2
 k 
 

2. Prove that the four points whose coordinates are (5, –1, 1), (7, –4, 7), (1, –6, 10), (–1, –3, 4) are the vertices of a rhombus.

  
3. Find the distance of the point whose spherical polar coordinates are  2 2 , ,  from the point whose Cartesian
 4 6

coordinates are 2 3 ,  1,  4 . 
Ans. 43

4. Show that the points A(1, 2, 3), B(4, 0, 4) and C(–2, 4, 2) are collinear.

5. Find the ratio in which the coordinate plane divide the line joining the points (–2, 4, 7), (3, –5, 8).

Ans. xy plane: –7 : 8; yz plane: 2 : 3; zx plane: 4 : 5

6. Find the direction cosine of a line that makes equal angles with the axes.

 1 1 1 
Ans.   , , 
 3 3 3

7. If A, B, C, D are the points (3, 4, 5), (4, 6, 3), (–1, 2, 4) and (1, 0, 5), find the projection of CD on AB.

Ans. 4/3

8. Show that the straight line whose dc's are given by the equation: ul + vm + wn = 0, al2 + bm2 + cn2 = 0 are

(i) perpendicular if u2(b + c) + v2(c + a) + w2(a + b) = 0

 u2   v2   w2 
(ii) parallel if    0
 a   b   c 
     

9. Prove that the straight line whose direction cosines are given by relations al + bm + cn = 0 and fmn + gnl + hlm = 0 are
f g h
perpendicular if    0 and parallel if af  bg   ch  0 .
a b c
10. If l1, m1, n1 and l2, m2, n2 are the dc's of two lines, then the direction ratio of another which is perpendicular to both
the given lines are (m1n2 – m2n1), (n1l2 – n2l1), (l1m2 – l2m1). Prove further if the given line are at right anlges to each
other then there dr's are the actual dc's.

4
11. A line makes angles  ,  ,  and  with the four diagonals of a cube. Prove that cos 2   cos 2   cos 2   cos 2   .
3

12. If two pairs of opposite edges of a tetrahedron are perpendicular, show that the third pair is also perpendicular.
 Analytic Geometry 7
13. Prove that three concurrent lines with direction cosines (l1, m1, n1), (l2, m2, n2) and (l3, m3, n3) are coplanar if

l1 m1 n1
l2 m2 n2  0 .
l3 m3 n3

14. A plane makes intercepts OA, OB, OC whose measures are a, b, c on the axes OX, OY, OZ. Find the area of the triangle
ABC.

1 2 2
Ans. a b  b2c 2  c 2 a2
2

15. (l1, m1, n1) and (l2, m2, n2) are the dc's of two concurrent lines, show that the dc's of two lines bisecting the angles
between them are proportional to (l1 ± l2), (m1 ± m2), (n1 ± n2).

16. The direction cosines of a variable line in two adjacent positions are l, m, n; l  l , m  m , n  n . Show that the small
angle  between the two positions is given by      l    m    n  .
2 2 2 2

PLANE
1. A variable plane moves such that the sum of reciprocals of its intercepts on the three coordinate axes is constant.
Show that it passes through a fixed point.

2. A plane meets the coordinate axes in A, B, C such that the centroid of the triangle ABC is the point (p, q, r). Show that
x y z
the equation of the plane is  p    q    r   3.
     

3. A plane makes intercepts –6, 3, 4 upon the coordinate axes. What is the length of perpendicular from the origin on it.

12
Ans.
29

4. Show that the four points (0, –1, 0), (2, 1, –1), (1, 1, 1) and (3, 3, 0) are coplanar.

5. Find the equation of the plane passing through the lines of intersection of the planes 2x – y = 0 and 3z – y = 0 and
perpendicular to the plane 4x + 5y – 3z = 8.

Ans. 28x – 17y + 9z = 0

6. Find the equation of the plane perpendicular to yz plane and passing through the points (1, –2, 4) and (3, –4, 5).

Ans. y + 2z – 6 = 0

7. Find the equation of the plane which passes through the point (–1, 3, 2) and is perpendicular to each of the two planes
x + 2y – 2z = 5 and 3x + 3y + 2z = 8.

Ans. 2x – 4y + 3z + 8

8. Find the distance between the parallel planes 2x – 2y + z + 3 = 0 and 4x – 4y + 2z + 5 = 0

Ans. 1/6

9. The sum of the distances of any number of fixed points from a plane is zero. Show that the plane always passes
through a fixed point.

10. Show that the plane 14x – 8y + 13 = 0 bisects the obtuse angle between planes 3x + 4y – 5z + 1 = 0 and 5x + 12y – 13z = 0.
8 Mathematics (Optional) Foundation Course for CSE

11. The plane lx + my = 0 is rotated through an angle  about its line of intersection with the plane z = 0. Prove that the
equation of the plane in its new position is lx  my  z l 2  m2 tan   0 .

3 4 5
12. Prove that    0 represents a pair of planes.
yz zx xy

13. Through a point P   , ,   a plane is drawn at right angles to OP to meet the axes in A, B, C. Prove that the area of the

p5
triangle ABC is 2  where OP = p.
 
14. A variable plane is at a constant distance p from the origin and meets the axes in A, B and C. Show that the locus of
the centroid of the tetrahedron OABC is x–2 + y–2 + z–2 = 16p–2.

15. A triangle, the length of whose sides are a, b and c is places so that the middle points of the sides are on the axes. Show
that the lengths  intercepted on the axes are given by 8 2  b 2  c 2  a 2 , 82  c 2  a2  b 2 , 8 2  a 2  b 2  c 2 .
Find the coordinates of its vertices.

Ans.   , ,   ,   ,  ,   ,   , ,   

STRAIGHT LINE-I
1. Find the ratio in which the join of (2, 3, 1) and (–2, 1, –3) is cut by the plane x – 2y + 3z + 4 = 0. Find also the coordinates
of the point of intersection.

2. Find the image of the point P(3, 5, 7) in the plane 2x + y + z = 6.

Ans. (–1, 3, 5)

x y z
3. Find the distance of the point (1, –2, 3) from the plane x – y + z = 5 measured parallel to the line  
2 3 6

Ans. 1

x y z x y z
4. Find the equation of the line through   , ,   at right angles to the lines   and  
l1 m1 n1 l2 m2 n2

x y  z
Ans. m n  m n  n l  n l  l m  l m
1 2 2 1 1 2 2 1 1 2 2 1

5. Find the incentre of the tetrahedron formed by the planes x = 0, y = 0, z = 0 and x + y + z = a.

6. P is a point on the plane lx + my + nz = p. A point Q is taken on the line OP such that OP.OQ = p2, prove that the locus
of Q is p(lx + my + nz) = x2 + y2 + z2.

7. A variable plane makes intercepts on the coordinate axes, the sum of whose squares is constant and equal to k2. Show
that the locus of the foot of the perpendicular from the origin to the plane is (x–2 + y–2 + z–2)(x2 + y2 + z2)2 = k2.

8. Find the equation of the line through the points (a, b, c) and (a', b', c') and prove that it passes through the origin, if
aa' + bb' + cc' = rr', where r and r' are the distances of these points from the origin.
 Analytic Geometry 9
9. Find the symmetric form of the equation of line given by x = ay + b, z = cy + d.

xb y0 zd

Ans.  
a 1 c

10. Find the symmetric form of the line 3x + 2y + z = 5, x + y – 2z = 3.

x1 y4 z
Ans.  
5 7 1

11. Find the equation to the plane through the points (2, –1, 0), (3, –4, 5) parallel to the line 2x = 3y = 4z.

Ans. 29x – 27y – 22z – 85 = 0

12. Find the equation of the plane through the line of intersection of the planes ax + by + cz + d = 0, a'x + b'y +c'z +d' = 0
and parallel to x-axis.

Ans. (ba' – ab')y + (ca' – c'a)z + (da' – d'a) = 0

x py  q rz  s
13. Prove that the plane through the point   , ,   and the line x = py + q = rz + s is given by  p  q r   s  0
1 1 1

14. Prove that the equation of the two planes inclined at an angle  to the xy plane and containing the line y = 0, z

2 2

2 2 2 2
cos   x sin  is x  y tan   z  2 zx tan   y tan  .

15. Find the equation of a system of planes perpendicular to the line with direction ratios, a, b, c.

Ans. ax + by + cz + k = 0

1 1 1
16. Find the foot and hence length of the perpendicular from (5, 7, 3) to the line x  15    y  29     z  5  . Find
3 8 5
also the equation of the plane in which the perpendicular and the given straight line lie.

17. Find the equations of the perpendicular from the origin to the line ax + by + cz + d = 0 = a'x + b'y + c'z + d'.

1 1 1
18. Find the distance of the point (3, 8, 2) from the line  x  1    y  3    z  2  measured parallel to the plane
2 4 3
3x + 2y – 2z + 15 = 0.

Ans. 7

x1 y2 z3 x2 y3 z4

19. Prove that the lines   and   are coplanar. Also find their point of intersection.
2 3 4 3 4 5

Ans. (–1, –1, –1)

20. Show that the lines x + y + z – 3 = 0 = 2x + 3y + 4z – 5 and 4x – y + 5z – 7 = 0 = 2x – 5y – z – 3 are coplanar and find the
plane in which they lie.

Ans. x + 2y + 3z = 2

x y z
21. Find the equation of the plane through the line   and perpendicular to the plane containing the lines
l m n
x y z x y z
  and   .
m n l n l m
Ans. (m – n)x + (n – l)y + (l – m)z = 0
10 Mathematics (Optional) Foundation Course for CSE

STRAIGHT LINE-II
1. Find the equation of the line which intersect the lines 2x + y – 4 = 0 = y + 2z and x + 3z = 4, 2x + 5z = 8 and passes
through the point (2, –1, 1).

Ans. x + y + z = 2, x + 2z = 4

y2 z5 x3 y5 z3

2. A line with DR's (7, –5, 2) is drawn to intersect the line x  7   ,   . Find the coordinates
1 1 3 2 4 3
of the points of intersection and the length intercepted on it.

3. Show that the equation of the line through (a, b, c) which is parallel to the plane lx + my + nz = 0 and intersects the line
A1x + B1y + C1z + D1 = 0 = A2x + B2y + C2z + D2 is

A1 x  B1 y  C 1 z  D1 A2 x  B2 y  C 2 z  D2
l  x  a   m  y  b   n  z  c   0, 
A1 a  B1 b  C1c  D1 A2 a  B2 b  C 2 c  D2

4. Show that the equation of the straight line through the origin cutting each of the lines

x y z x y z
x  x1 y  y1 z  z1 x  x 2 y  y2 z  z2 z1  x2 z2  0
  and   is x1 y1 y2
l1 m1 n1 l2 m2 n2 l1 m1 n1 l2 m2 n2

INTERSECTION OF THREE PLANES

5. Examine the nature of intersection of planes: 2x – y + z = 4, 5x + 7y + 2z = 0, 3x + 4y – 2z + 3 = 0

Ans. Point (1, –1, 1)

6. Show that the planes 2x + 4y + 2z = 7, 5x + y – z = 9; x – y – z = 6 form a triangular prism.

7. Prove that the planes 2x – 3y – 7z = 0, 2x – 14y – 13z = 0, 8x – 31y – 33z = 0 pass through one line and find its equation.

8. Prove that the planes x = cy + bz, y = az + cx, z = bx + ay pass through one line if a2 + b2 + c2 + 2abc = 1 and find its
equations.

x y z
Ans.  
2 2
1 a 1b 1  c2

9. For what value of  do the planes x  y  z  1  0, x  3y  2 z  3  0, 3x  y  z  2  0

(i) Intersect in a point

(ii) Intersect along a line

(iii) Form a triangular prism

Ans. (i) Point:   4,   3 ; Line:   3 ; (iii)   4

10. How far is the point (4, 1, 1) from the line of intersection of x + y + z – 4 = 0 = x – 2y – z – 4.

3
Ans. 42
14
 Analytic Geometry 11
x1 y 3 z1
11. Find the equation of the two planes through the origin which are parallel to the line   are distance
2 1 2
5
from it.
3

Ans. x – 2y + 2z = 0 and 2x + 2y + z = 0

12. Find the length and equations of the perpendicular from the origin to the line x + 2y + 3z + 4 = 0 = 2x + 3y + 4z + 5. Also
find the coordinates of the foot of the perpendicular.

21  2 1 4  x y z
Ans. , , , ,  
3  3 3 3  2 1 4

13. Find the equation of the right circular cylinder of radius 2 whose axis passes through (1, 2, 3) and has direction
cosines proportional to (2, 3, 6).

Ans. 196 = (3x + 2y – 7)2 + 9(2y – z – 7)2 + 4(3x – z)2

SKEW LINES
x3 y5 z2 x1 y1 z1
14. Find the SD between the lines   and   .
1 2 1 7 6 1

34
Ans.
29

x3 y1 z x 2 y z7

15. Find the length and the equation common perpendicular to the two lines   and   .
4 3 2 4 1 1

 32 x  34 y  13 z  108  0 
Ans. 9,  
 4x  11y  5z  27  0 

x3 y8 z3 x3 y7 z6

16. Find the SD between the lines   and   . Find also its equation and the points
3 1 1 3 2 4
where it meets the given lines.

x3 y8 z3

Ans. 3 30 ,  
2 5 1

17. Show that the SD between any two opposite edges of the tetrahedron formed by the planes y + z = 0, z + x = 0,
2a
x + y = 0, x + y + z = a is and the three lines of SD intersect at the point x = y = z = –a.
6

x   1 y  1 z   1 x   2 y   2 z   2
Two straight lines   ,   , ,  .
18. l1 m1 n1 l2 m2 n2 are cut by a third line whose dc's are

l1 m1 n1 1   2 1   2 1  2
Show that 'd' the length intercepted on the third line is given by d l2 m2 n2  l1 m1 n1 . Deduce
   l2 m2 n2
the length of SD between the first two lines.

1   2 1   2 1   2
l1 m1 n1
l2 m2 n2
Ans. SD  d 
 m n1 2  m2 n1 
2

12 Mathematics (Optional) Foundation Course for CSE

STRAIGHT LINE
1. Prove that the locus of a variable line which intersect the three given lines y = mx, z = c; y = –mx, z = –c; y = z, mx = –c
is the surface y2 – m2x2 = z2 – c2.

2. Find the surface generated by a line which intersects the line y = a = z and x + 3z = a = y + z and is parallel to the plane
x + y = 0.

3. Find the surface generated by a straight line which intersects the line x + y = 0 = z, x – y – z = 0 = x + y – 2a and the
parabola y = 0 = x2 – 2az.

4. Prove that the locus of a line which meets the lines y = ±mx, z = ±c and the circle x2 + y2 = a2, z = 0 is

c2m2(cy – mxz)2 + c2(yz – cmx)2 = a2m2(z2 – c2)2

 x2   y2 
5. A straight line is drawn through a variable point on the ellipse  2
a    2   1, z = 0 to meet two fixed line y = mx,
  b 
z = c and y = –mx, z = –c. Find the locus of the straight line.

Ans. (cmx – yz)2c2b2 + (mxz – cy)2c2a2m2 = a2b2m2(z2 – c2)2

SPHERE
1. Find the centre and the radius of the sphere x2 + y2 + z2 – 2x + 4y – 6z = 11.

Ans. (1, –2, 3), radius = 3

2. Find the equation of the sphere which passes through (a, 0, 0), (0, b, 0), (0, 0, c) and (0, 0, 0).

Ans. x2 + y2 + z2 – ax – by – cz = 0

3. Obtain the equation of sphere having its centre on the line 5x + 2z = 0 = 2x – 3y and passing through the points
(0, –2, –4) and (2, –1, –1).

Ans. x2 + y2 + z2 – 6x – 4y – 10z + 12 = 0

4. A sphere of radius 'k' passes through the origin and meets the axes in A, B, C. Prove that the centroid of the triangle ABC
lies on the sphere 9(x2 + y2 + z2) = 4k2.

5. A plane passes through a fixed point (p, q, r) and cuts the axes in A, B, C, show that the locus of the centre of the sphere
OABC is

p q r
  =2
x y z

6. Find the equation of the sphere that passes through the points (4, 1, 0), (2, –3, 4), (1, 0, 0) and touches the plane
2x + 2y – z = 11.

Ans. x2 + y2 + z2 – 6x + 2y – 4z + 5 = 0

7. A sphere of constant radius 2k passes through the origin and meets the axes in A, B, C. Find the locus of the centroid of
the tetrahedron OABC.

Ans. x2 + y2 + z2 = k2

8. Find the equation of the sphere which passes through the points (1, 0, 0), (0, 1, 0) and (0, 0, 1) and has its radius as small
as possible.
 Analytic Geometry 13
9. OA, OB, OC are three mutually perpendicular lines through the origin having direction cosines l1, m1, n1 ; l2, m2, n2 and
l3, m3, n3. If OA = a, OB = b, OC = c. Find the equation of sphere OABC.

(10) Find the radius and centre of the circle x2 + y2 +z2 – 8x + 4y + 8z – 45 = 0, x – 2y + 2z = 3

 13 8 10 
Ans. 4 5,  , , 
 3 3 3 

11. Find the equation of the sphere whose centre is the point (1, 2, 3) and which touches the plane 3x + 2y + z + 4 = 0. Find
also the radius of the circle in which the sphere is cut by the plane x + y + z = 0.

Ans. x2 + y2 + z2 – 2x – 4y – 6z = 0

12. Find the equation of the sphere through the circle x2 + y2 + z2 = 9, x + y – 2z + 4 = 0 and the origin.

Ans. 4x2 + 4y2 + 4z2 + 9x + 9y – 18z = 0

13. Prove that the plane x + 2y – z = 4 cuts the sphere x2 + y2 + z2 – x + z + 2 = 0 in a circle of radius unity and find the
equations of the sphere which has this circle for one of its great circles.

Ans. x2 + y2 + z2 – 2x – 2y + 2 = 0

14. Prove that the circle x2 + y2 + z2 – 2x + 3y + 4z – 5 = 0, 5y + 6z + 1 = 0 and x2 + y2 + z2 – 3x – 4y + 5z – 6 = 0, x + 2y – 7z =

0 lies on the same sphere and find its equation. Also find the value of 'a' for which x + y + z = a 3 touches the sphere.

Ans. a 3 3

15. Find the equations of the sphere which pass through circle x2 + y2 + z2 = 5, x + 2y + 3z = 3 and touch the plane
4x + 3y = 15.

4 8 12 13
Ans. x  y  z  x y z
2 2 2
=0
5 5 5 5

16. P is the variable point on the given line and A, B, C are its projections on the axes. Show that the sphere O, ABC passes
through a fixed circle.

Ans. x2 + y2 + z2 – x – y – z = 0, lx + my + nz = 0.

x y z
17. A variable is parallel to the given plane   = 0 and meets the axes in A, B, C respectively. Prove that the circle ABC
a b c

b c c a a b
lies on the cone yz     zx     xy    = 0
c b a c b a

18. Find the equation of the sphere which passes through the point (, , ) and the circle x2 + y2 = a2, z = 0.

19. Find the plane, the centre and the radius of the circle common to the two spheres x2 + y2 + z2 – 4z + 1 = 0 and x2 + y2 +
z2 – 4x – 2y – 1 = 0

2 1 4 1
Ans. 2x + y – 2z + 1 = 0,  , ,  ,
3 3 3 3

x2 y 2
20. POP' is a variable diameter of the ellipse z = 0,  = 1 and a circle is described in the plane PP' zz' on PP' as
a2 b2
 x2 y2 
diameter, prove that as PP' varies the circle generates the surface  x  y 2  z 2   2  2  = x2 + y2 .
2

a b 
14 Mathematics (Optional) Foundation Course for CSE

21. A sphere whose center lies in the positive octant passes through the origin and cuts the planes x = 0, y = 0, z = 0 in circles

of radii a 2 , b 2 , c 2 respectively. Find the equation of this sphere.

Ans. x 2  y 2  z 2  2 x  b 2  c 2  a 2   2 y c 2  a 2  b 2  2 z a 2  b 2  c 2 = 0

22. A is point on OX and B on OY, so that the angle OAB is constant and equal to . On AB as diameter a circle is drawn
whose plane is parallel to OZ. Prove that as AB varies the circle generates the cone 2xy – z2sin2 = 0.

23. Sphere are described to contain the circle z = 0, x2 + y2 = a2. Prove that the locus of the extremities of their diameters
which are parallel to the x-axis is the rectangular hyperbola x2 – z2 = a2, y = 0.

Tangent Planes

24. Show that the plane 2x + y – z = 12 touches the sphere x2 + y2 + z2 = 24 and find its point of contact.

Ans. (4, 2, –2)

25. Find the equation of the tangent planes to the sphere x2 + y2 + z2 – 4x + 2y – 6z + 5 = 0, which are parallel to the plane
2x + y – z = 0.

Ans. 2 x  y  z  3 6  0

26. If three mutually perpendicular chords of lengths d1, d2, d3 be drawn through the point (, , ) to the sphere x2 + y2 +
z2 = a2, prove that d12  d22  d32 = 12a2 – 8(2 +  2 + 2).

27. Find the equations of the tangent line to the circle 3x2 + 3y2 + 3z2 – 2x – 3y – 4z – 22 = 0, 3x + 4y + 5z – 26 = 0 at the point
(1, 2, 3).

28. Find the equation of a sphere touching the three coordinate planes. How many such spheres can be drawn.

29. A sphere touches the three coordinate planes and passes through the point (2, 1, 5). Find its equation.

Ans. x2 + y2 + z2 – 10(x + y + z) + 30 = 0

30. Prove that the centres of the spheres which touch the lines y = mx, z = c, y = –mx, z = –c lie upon the conicoid
mxy + cz(1 + m2) = 0

31. Find the locus of the centres of spheres of spheres of constant radius which pass through a given point and touch a
given line.

Ans. x2 – 2az + a2 = 0 and y2 + z2 = k2

32. Find the locus of the centres of spheres which pass through a given point and touch a given plane.

Ans. x2 + y2 – 2az + a2 = 0
 Analytic Geometry 15

Touching Sphere
33. Show that the spheres x2 + y2 + z2 = 100 and x2 + y2 + z2 – 24x – 30y – 32z + 400 = 0 touch externally and find their point
of contact.

 24 32 
Ans.  , 6, 
 5 5 

34. Show that the spheres x2 + y2 + z2 = 64 and x2 + y2 + z2 – 12x + 4y – 6z + 48 = 0 touch internally and find their point of
contact.

 48 1 24 
Ans.  , , 
 7 7 7 

Pole/Polar
35. Find the pole of the plane lx + my + nz = p w.r.t. the sphere x2 + y2 + z2 = a2.

 a2 l a2 m a2 n 
Ans.  p , p , p 
 

x y1 z3
36. Prove that the polar plane of any point on the line   with respect to the sphere x2 + y2 + z2 = 1 passes
2 3 4

 1   1 
through the line    2 x  3  =   ( y  1) = –z.
 13   3 

Angle of Intersection
37. Show that the two spheres x2 + y2 + z2 + 6y + 2z + 8 = 0 and x2 + y2 + z2 + 6x + 8y + 4z + 20 = 0 are orthogonal. Find their
plane of intersection.

Ans. 3x + y + z + 6 = 0.

38. Two points P and Q are conjugate with respect to a sphere S; prove that the sphere on PQ as diameter cuts S orthogonally.

39. Find the equation of the sphere which touches the plane 3x + 2y – z + 2 = 0 at the point (1, –2, 1) and cuts orthogonally
the sphere x2 + y2 + z2 – 4x + 6y + 4 = 0

Ans. x2 + y2 + z2 + 7x + 10y – 5z + 12 = 0.

r1 r2
40. Two spheres of radii r1 and r2 cut orthogonally. Prove that the radius of the common circle is .
 r12  r22 
41. Find the equation of a sphere which cuts the four given spheres orthogonally.

x 2  y 2  z2 x y z 1
d1 u1 v1 w1 1
d2 u2 v2 w2 1  0
Ans. d3 u3 v3 w3 1
d4 u4 v4 w4 1
16 Mathematics (Optional) Foundation Course for CSE

42. Find the length of the tangent drawn from the point (1, 2, 3) to the sphere

5(x2 + y2 + z2) – x + 10y + 20z + 8 = 0

157
Ans. PT =
5

Coaxial System
43. Prove that every sphere that passes through the limiting points of a coaxial system cuts every sphere of that system
orthogonally.

44. Find the limiting points of coaxial systems defined by the spheres

x2 + y2 + z2 + 2x + 2y + 4z + 2 = 0 and x2 + y2 + z2 + x + y + 2z + 2 = 0

 1 1 2   1 1 2 
Ans.  , ,  and   , , 
 3 3 3  3 3 3

CONE
1. Find the equation of the cone whose vertex is at the origin and base is the circle x = a, y2 + z2 = b2 and show that the
section of the cone by a lane parallel to the plane X-Y is a hyperbola.

Ans. b2x2 + a2y2 – a2z2 = 0

x y z
2. The plane    1 meets the coordinate axes is A, B, C. Prove that the equation of the cone generated by lines
a b c
drawn from O to meet the circle ABC is

b c c a a b
yz     zx     xy    = 0
c b a c b a

3. Planes through OX, OY include on angle .

Show that their line of intersection lies on the cone z2(x2 + y2 + z2) = x2y2tan2.

x y z x y z
4. Find the equation of the cone which passes through three coordinate axes and the lines   ;  
1 2 3 3 2 1

Ans. 3yz + 10zx + 6xy = 0

5. OP and OQ are two lines which remain perpendicular and move so that the plane OPQ passes through OZ. If OP

y z  y  x y 2  
describes the cone f  ,  = 0, prove that OQ describes the cone  x ,   z  zx   0
f
x x  

6. Find the equation of a conc whose vertex is (, , ) and base y2 = 4ax, z = 0.

Ans. (z – y)2 = 4a(z – x) (z – )

7. A cone has as base the circle x2 + y2 + 2ax + 2by = 0, z = 0 and passes through the fixed point (0, 0, c). If the section of the
cone by zx plane is a rectangular hyperbola, prove that the vertex lies on fixed circle.
 Analytic Geometry 17
8. Prove that the equation 4x2 – y2 + 2z2 + 2xy – 3yz + 12x – 11y + 6z + 4 = 0 represents a cone. Hence find its vertex.

Ans. (–1, –2, –3)

 
9. Prove that the angle between the lines given by x + y + z = 0, ayz + bzx + cxy = 0 is if a + b + c = 0 and if
2 3
1 1 1
  = 0.
a b c
10. If the plane 2x – y + cz = 0 cuts the cone yz + zx + xy = 0 in perpendicular lines, find the value of 'c'.

Ans. 2

x y z
11. If   represent one of a set of three mutually perpendicular generators of the cone 5yz – 8zx – 3xy = 0, find the
1 2 3
equations of the other two.
x y z x y z
Ans.   &  
1 1 1 5 4 1
12. Show that the locus of points from which three mutually perpendicular lines can be drawn to intersect a given circle
x2 + y2 = a2, z = 0 is a surface of revolution.

Ans. x2 + y2 + 2z2 = a2

13. Find the locus of points from which three mutually perpendicular lines can be drawn to intersect the conic z = 0,
ax2 + by2 = 1

Ans. ax2 + by2 + (a + b)z2 = 1

x 2 y 2 z2
14. Three points P, Q, R are taken on the ellipsoid   = 1. So that line joining P, Q, R to the origin are mutually
a2 b2 c 2
perpendicular. Prove that the plane PQR touches a fixed sphere.

Ans. x2 + y2 + z2 = 2

x 2 y 2 z2
15. Prove that the cones ax2 + by2 + cz2 = 0 and   = 0 are reciprocal to each other.
a b c

16. A line OP is such that the two planes through OP each of which cuts the cone ax2 + by2 + cz2 = 0 in perpendicular
generators are perpendicular, prove that the locus of OP is a cone and find it.

Ans. (2a + b + c)x2 + (2b + c + a)y2 (2c + a + b)z2 = 0

17. Show that the general equation to a cone which touches the coordinate planes is a2x2 + b2y2 + c2z2 – 2bcyz – 2cazx – 2abxy
=0

18. Prove that the tangent lines from the origin of coordinates to the sphere (x – a)2 + (y – b)2 + (z – c)2 = k2 lie on the cone given
by the equation (a2 + b2 + c2 – k2) (x2 + y2 + z2) = (ax + by + cz)2 .

19. Show that the three mutually perpendicular tangent lines can be drawn to the sphere x2 + y2 + z2 = r2 from any point on

2 2 2 3 2
the sphere x  y  z  r .
2

20. Find the equation to the right circular cone whose vertex is (2, –3, 5), axis makes equal angles with the coordinate axes
and semi vertical angle is 30°.

Ans. 5(x2 + y2 + z2) – 8(xy + yz + zx) – 4x + 86y – 58z + 278 = 0

21. Find the equation of the cone formed by rotating the line 2x + 3y = 6, z = 0 about the y axis.
18 Mathematics (Optional) Foundation Course for CSE

CYLINDER
1. Find the equation of the cylinder with generators parallel to z-axis and passing through the curve ax2 + by2 = 2cx,
lx + my + nz = p.
2. Find the equation of the surface generated by a straight line which is parallel to the line y = mx, z = nx and intersect
x2 y2
the ellipse   1, z  0.
a2 b 2

b 2  nx  z   a 2  ny  mz   a 2 b 2 n 2
2 2
Ans.

3. Find the equation of right circular cylinder whose axis is x = 2y = –z and radius is 4.

Ans. 5x 2  8 y 2  5 z 2  4 yz  8xz  4 xy  144

4. Find the equation of right circular cylinder whose axis is x – 2 = z, y = 0 and passes through the point (3, 0, 0).

Ans. x 2  2 y 2  z 2  2 zx  4 x  2 z  3  0

5. Find the equation of the right circular cylinder which passes the circle x2 + y2 + z2 = 9, x – y + z = 3.

Ans. x 2  y 2  z 2  xy  xz  yz  9  0

6. Show that the equation of the right circular cylinder described on the circle through the three points A(1, 0, 0), B(0,
1, 0) and C(0, 0, 1) as the guiding curve is x2 + y2 + z2 – yz – zx – xy = 1.
7. Find the equation of the enveloping cylinder of the sphere x2 + y2 + z2 – 2x + 4y = 1 whose generators are parallel
to the line x = y = z.

Ans. x 2  y 2  z 2  yz  zx  xy  4 x  5 y  z  2  0

8. Show that the enveloping cylinder of the conicoid ax2 + by2 + cz2 = 1 with generators perpendicular to z-axis meets
the plane z = 0 in parabolas.

ab  mx  ly   ab 2  bm 2 , z  0
2
Ans.

x 2 y 2 z2
9. Find the equation of the enveloping cone of the ellipsoid    1 and deduce from it the equation of the
a2 b 2 c 2
x y z
enveloping cylinder whose generators are parallel to the line   .
l m n
2
 x2  l 2   lx my nz 
Ans.  2  1   2  2  2  
 a  a  a b cz 

10. Find the equation of the enveloping cylinder of the ellipsoid ax2 + by2 + cz2 = 1 whose generators are parallel to the
line x = y = z.

Ans.  b  c  x 2   c  a  y 2   a  b  z2  2 abxy  2bcyz  2cazx   a  b  c   0

CONICOID
1. Find the equation of the tangent planes to the hyperboloid 2x2 – 6y2 + 3z2 = 5 which pass through the line
x + 9y – 3z =0 = 3x –3y + 6z – 5.

2. Tangent planes are drawn to the conicoid ax2 + by2 + cz2 = 1 through (). Show that the perpendicular from the
centre to the conicoid to these planes generate the cone.

x 2 y 2 z2
 x  y  z 
2
  
a b c
 Analytic Geometry 19
x 2 y 2 z2
3. A tangent plane to the ellipsoid    1 meets the coordinate axis in the points P, Q and R. Find the locus
a2 b 2 c 2
of the centroid of the triangle PQR.

x 2 y 2 z2
4. Find the locus of the foot of the central perpendicular on varying tangent planes to the ellipsoid    1.
a2 b 2 c 2
x 2 y 2 z2
5. If 2r is the distance between the parallel tangent planes to the ellipsoid    1 , prove that a line through
a2 b 2 c 2
the origin perpendicular to the planes lies on the cone x  a  r   y  b  r   z  c  r   0.
2 2 2 2 2 2 2 2 2

6. Show that the tangent planes at the extremities of any diameter of an ellipsoid are parallel.

7. Through a fixed point (k, 0, 0) pairs of perpendicular lines are drawn to the conicoid ax 2  by 2  cz 2  1. Show that

x  k
2
y2 z2
the planes through any pair touches the cone    0.
 b  c   ak 2  1  c  ak 2  1   a b  ak 2  1   a
8. Find the surface generated by straight lines drawn through a fixed point () at right angles to their polar with
respect to the conicoid ax 2  by 2  cz 2  1.

9. Find the locus of straight lines through a fixed point () whose polar lines with respect to the quadrics
ax 2  by 2  cz 2  1 and ax 2  by 2  cz 2  1 are coplanar.

10. Prove that the centres of sections of the ax 2  by 2  cz 2  1, by the planes which are at a constant distance p from

the origin lie on the surface  ax  by  cz   p  a x  b y  c z 

2 2 2 2 2 2 2 2 2 2

11. Show that a line joining a point P to the centre of a conicoid ax 2  by 2  cz 2  1 passes through the centre of the
section of the conicoid by the polar plane of P.

12. Find the locus of the centres of the sections ax 2  by 2  cz 2  1 which touches x 2   y 2  z 2  1.

13. Prove that the middle points of the chords of ax 2  by 2  cz 2  1, which are parallel to x = 0 and touch x2 + y2 + z2 = r2
lie on the surface by2 (bx2 + by2 + cz2 – br2) + cz2 (cx2 + by2 + cz2 – cr2) = 0.
x2
14. Find the length of the normal chord through P of the ellipsoid  2  1 and prove that if it is equal to 4PG3, where
a
G 3 is the point where the normal chord though P meets the XY plane, then P lies on the cone

x2 y2 z2
6 
2c 2  a 2   6  2c 2  b 2   4  0.
a b c

 x2 
15. The normal at a variable point P of the ellipsoid  a 2 
 1 meets the xy plane in G3 and G3Q is drawn parallel
 
x2 y2 z2
to z-axis and equal to G3P. Prove that the locus of Q is given by 2 2
 2 2  2  1. Find the locus of R, if OR
a c b c c
is drawn from the centre equal and parallel to G3P.

 x2 
16. Normals at P and P, points of the ellipsoid  a
2 
 1 , meet the xy plane in G2 and G3 and make angles  and 
 
with PP. Prove that PG3 cos  + PG3 cos  = 0.
17. Prove that the lines drawn from the origin parallel to the normal of ax2 + by2 + cz2 = 1 at its point of intersection with
the plane lx + my + nz = p generate the cone.
2
 x 2 y 2 z 2   lx my nz 
p2       
 a b c   a b c 
20 Mathematics (Optional) Foundation Course for CSE

x2
18. If P, Q, R, P, Q, R are the feet of the six normals from a point to the ellipsoid a 2
 1 , and the plane PQR is given

x y z 1
by lx + my + nz = p, prove that the plane PQR is given by 2
 2  2   0.
al b m c n p

x2
19. If OP, OQ and OR be the conjugate semi-diameters of the ellipsoid  a2  1 and P, Q, R be (x1, y1, z1), (x2, y2, z2) and
(x3, y3, z3) respectively, then

(i) Find the equation of the plane PQR.

(ii) Prove that if the plane lx + my + nz = p, passes through the points P, Q, R then a2l2 + b2m2 + c2n2 = 3p2.

(iii) Prove that the pole of the plane PQR lies on the ellipsoid.

x2
20. If the axes are rectangular, find the locus of the equal conjugate diameters of the ellipsoid a 2
 1.

x2 x2 1
21. Prove that the locus of the section of the ellipsoid a 2
 1 by the plane PQR is the ellipsoid a 2
 .
3

22. Find locus of the asymptotic line drawn from the origin to the conicoid ax2 + by2 + cz2 = 1.

PARABOLOID

 x2   y2 
1. Show that the plane 8x – 6y – z = 5 touches the paraboloid       z , and find the point of contact.
 2   3 

Ans. (8, 9, 5)

x2 y2  z  x2 y2  z  x2 y 2 2z
2. Find the condition that a 2  b 2  2  c  , a 2  b 2  2  c  ; 2  2  have a common tangent plane.
1 1  1 2 2  2  a3 b3 c 3

a12 b12 c1
Ans. a22 b22 c2  0
a32 b32 c3

x2 y 2
3. Two perpendicular tangent planes to the paraboloid   2 z intersect in a line lying on the plane x = 0.
a b
Prove that the line touches the parabola x = 0, y2 = (a + b) (2z + a).
4. Find the equation of the plane which cuts the paraboloid x2 – 2y2 = 3z in the conic with centre (1, 2, 3).
Ans. 2x – 8y – 3z + 23 = 0
5. Show that the feet of the normals from the point () on the paraboloid x2 + y2 = 2az lie on a sphere.

   2  2  
Ans. x  y  z     a z  
2 2 2
y  0
2 
 

6. Prove that the equations of the chord through the point (1, 2, 3) which is bisected by the diametral plane
1 1
10x  24 y  21 of the paraboloid 5x 2  6 y 2  7 z are  x  1    y  2    z  3  .
2 3

7. Find the locus of the point from which three mutually perpendicular tangents can be drawn to the paraboloid.
 Analytic Geometry 21

GENERATING LINES

 x 2   y 2   z2 
1. Find the equations of the generators of the hyperboloid  2    2    2   1 which pass through the point
a  b  c 
(a cos , b sin , 0).

x  a cos  y  b sin  z
Ans.    learn this result 
a sin  b cos  c

x2 y 2
2. CP, CQ are any two conjugate semi-diameters of the ellipse   1, z  c , CP, CQ are the conjugate diameters
a2 b 2
 x2   y 2 
of the ellipse  2    2   1, z  c , drawn in the same directions as CP and CQ. Prove that the hyperboloid
a  b 
 2x   2 y  z2
2 2

    2   2  1 is generated by either PQ or PQ.

 a   b  c

x 2 y 2 z2
3. Prove that in general two generators of the hyperboloids    1 can be drawn to cut a given generator at
a2 b 2 c 2
right angles.

4. Find the locus of the point of intersection of perpendicular generators of a hyperboloid of one sheet.

Ans. x 2  y 2  z2  a 2  b 2  c 2

5. If A and A are the extremities of the major axis of the principal elliptic section and any generator meets the two
generators of the same system through A and A in P and P respectively, then prove that AP . AP = b2 + c2.

6. Show that the equations y  z    1  0,    1  x  y    0 represent for different values of  generators of one

system of the hyperboloid yz  zx  xy  1  0 and find the equations to the generators of the other system.

7. Find the locus of the point of intersection of perpendicular generators of the hyperbolic paraboloid.

Ans.  a 2  b 2  2 z  0 

8. Planes are drawn through the origin O and the generators through any point P of the paraboloid x2 – y2 = az. Prove
 2r 
that the angle between them is tan 1   , where 'r' is the length of 'OP'.
 a 
x2
9. Find the vertices of the skew quadrilateral formed by the four generators of the hyperboloid  y 2  z 2  49
4
passing through (10, 5, 1) and (14, 2, – 2).

