PART # 03

CO-ORDINATE GEOMETRY
EXERCISE # 01

SECTION-1 : (ONE OPTION CORRECT TYPE)

531. The co-ordinates of the point on the parabola y2 = 8x, which is at minimum distance from the circle
x2 + (y + 6)2 = 1 are
(A) (– 2, 4) (B) (2, – 4) (C) (2, 4) (D) none of these

532. The values of k for which the circles x2 + y2 = 1 and x2 + y2 – 4x + 8 = 0 have two common tangents is
 9 9  9 9 
(A)  ,  (B)  ,     ,  
 4 4  4 4 
 9 9 
(C)  ,  4    4 ,   (D) None of these
   

2 2 2 2
533. The chords of the hyperbola x  y = a touches parabola y = 4ax, then the locus of their mid-point is
(A) y2 (a  x) = x3 (B) x2 (a + x) = y3 (C) y2 (x  a) = x3 (D) y2 (x + a) = x3

534. Consider a triangle ABC, where B and C are (– a, 0) and (a, 0) respectively. A be any point (h, k). P, Q, R
divides the sides of this triangle in same ratio. Then centroid of PQR is
h   k h k
(A) (0, 0) (B)  3 , 0 (C)  0, 3  (D)  3, 3 
     

535. If a focal chord of the parabola y2 = 4ax be at a distance d from the vertex, then its length is equal to
a2 d2 4a3 2a 2
(A) (B) (C) (D)
d2 a d2 d

536. An ellipse slides between two perpendicular straight lines, then the locus of its centre is
(A) circle (B) parabola (C) ellipse (D) hyperbola

537. If the normal to the parabola y2 = 4ax at the point P(at2, 2at) cuts the parabola again at Q(aT2, 2aT), then
(A) 2T2 (B) T  ( ,  8)  (8, )
2 2
(C) T <8 (D) T 8

538. If the tangents at P and Q on a parabola meet in R and S be its focus. If p = SP, q = SR and r = SQ, the roots
of the equation px2 + 2qx + r = 0 are
(A) rational (B) real and equal (C) imaginary (D) real and unequal

539. Equation of the common tangent to the curves y2 = 8x and xy =  1 is

(A) 3y = 9x + 2 (B) y = 2x + 1
(C) 2y = x + 8 (D) y = x + 2

x2 y2 2 2 2
540. The tangent drawn from (, ) to an ellipse   1 touches the circle x + y = c , then the locus of (, )
a2 b2
is
(A) an ellipse (B) a circle (C) a parabola (D) none of these
541. Consider a point P(at2, 2at) on the parabola y2 = 4ax. A focal chord PS (S being focus) is drawn to meet
parabola again at Q. From Q, a normal is drawn to meet parabola again at R. From R, a tangent is drawn to
the parabola to meet focal chord PSQ (extended) at T. The area of QRT is
3 4 3
 1  1 8a2  1 
(A) 8a2  t 2  2  (B) a2  t   (C) t   (D) None of these
 t   t 3  t

542. Two pair of straight lines have the equations y2 + xy – 20 = 0 and ax2 + 2hxy + by2 = 0. One line will be
common among them if
(A) a = 8(h – 2b)or a = –5(2h + 5b) (B) a = 4(h – 2b) or a = –3(2h + 5b)
(C) a = 8(h – 2b) or a = 5(2h + 5b) (D) None of these

543. Let PQ, RS are the tangents at the extremities of a diameters PR of a circle of radius r such that PS, RQ
intersect at a point X on the circumference of the circle, then 2r equals
PQ  RS 2PQ  RS PQ2  RS2
(A) (B) PQ  RS (C) (D)
2 PQ  RS 2

x2 y2
544. If PQ is a double ordinate of the hyperbola   1 such that OPQ is an equilateral triangle. O being the
a2 b2
centre of the hyperbola, then the eccentricity e of the hyperbola satisfies
2 2 3 2
(A) 1<e< (B) e= (C) e= (D) e>
3 3 2 3

545. An equilateral triangle is inscribed in the circle x2 + y2 = a2 with vertex (a, 0) the equation of the side opposite
to the vertex is
(A) 2x  a = 0 (B) x+a=0
(C) 2x + a = 0 (D) 3x  2a = 0

546. If the normal at any point P on an ellipse meets major and minor axis at G and G and OF be the
perpendicular drawn from centre O to this normal then PF. PG must be equal to
(A) b2 (B) a2
(C) ab (D) None of these

547. The diameter of the smallest circle which touches the line y = 3x  3 and passes through a point on the
parabola y = x2 + 7x + 2
1 1 1 1
(A) (B) (C) (D)
2 5 10 2 10
548. If four distinct points of the curve y = 2x4 + 7x3 + 3x – 5 are collinear, then the A.M. of the x-co-ordinate of the
four points is
7 3 7 3
(A) – (B) (C) (D) –
8 4 8 4

549. Given a fixed circle C and a line L through the centre O of C. Take a variable point P on L and let K be the
circle centre P through O. Let T be the point where a common tangent to C and K meets K. The locus of T is
(A) a circle (B) a parabola
(C) a pair of straight lines (D) None of these
550. If an ellipse slides between two perpendicular straight lines, then the point of intersection of these two lines
lies on
(A) auxiliary circle of ellipse
(B) director circle of ellipse
(C) a line passes through centre of ellipse and perpendicular to axis of ellipse
(D) none of these

x2 y2
551. If a variable straight line x cos + y sin = p which is a chord of the hyperbola   1 (b > a) subtends a
a2 b2
right angle at the centre of the hyperbola then it always touches a fixed circle whose radius is
a2b2 ab ab ab
(A) (B) (C) (D)
2
b a 2 2
a b 2 2
b a 2 b  a2
2

552. Equation of the circle of which the points (1, 2) and (2, 3) are the ends of a chord of segment containing an
angle 450 is
2 2 2 2
(A) x + y – 4x – 4y + 7 = 0 (B) x + y – 4x – 4y – 14 = 0
2 2
(C) x + y + 4x + 4y + 7 = 0 (D) none of these
2
553. The locus of the mid point of the line segment joining the focus to a moving point on the parabola y = 4ax is another
parabola with directrix
a a
(A) x=–a (B) x= (C) x=– (D) x=0
2 2

2 2 2 2 2
554. If the two circles (x  1) + (y  3) = r and x + y  8x + 2y + 8 = 0 intersect in two distinct points, then
(A) r<2 (B) r=2 (C) r>2 (D) 2<r<8

555. A line 3x  4y + 4 = 0 is tangent to a circle whose radius is 3/4. If another straight line 3x  4y +  = 0 is also
tangent to same circle, then value of  is
3 7 4 2
(A) (B)  (C)  (D) 
4 2 3 7

SECTION–2 : (MORE THAN ONE OPTION CORRECT TYPE)

556. If the circle x2 + y2 – 9 = 0 and x2 + y2 + 2x + 2y + 1 = 0 touch each other, then  is

4 4
(A) – (B) 0 (C) 1 (D)
3 3

x2 y2 2
557. If the ellipse  y 2 = 1 meets the ellipse x 2  2  1 in four distinct points and a = b – 5b + 7 then b can
4 a
take values
(A) (– , 0) (B) [4, 5]
(C) [2, 3] (D) (0, )

558. The straight line x  y  0, 3 x  y  4  0 and x  3 y  4  0 form a triangle which is

(A) isosceles (B) right -angled
(C) obtuse - angled (D) equilateral
559. The point P(,  + 1) will lie inside the ABC whose vertices are A(0, 3), B(2, 0) and C(6, 1) if
1 1 6 3
(A)  = 1 (B)  (C)  (D)  
2 2 7 2

x2 y2 2 2
560. Tangents are drawn from any point with eccentric angle  on the hyperbola   1 to the circle x + y =
16 9
16. If (x1, y1) is midpoint of chord of contact, then  is equal to
4x1 16x12
(A) sec 1 (B) sec 1
x 2
1  y12  x 2
1  y12 
16y1 4y12
(C) tan1 (D) tan1

3 x12  y12  x 2
1  y12 
2
561. Consider a circle with its centre lying on the focus of the parabola y = 2px such that it touches the directrix of
the parabola, then a point of intersection of the circle and the parabola is
p  p  p  p 
(A)  , p (B)  ,  p (C)  , p (D)  ,  p
2  2  4  4 

x y
562. The equation of the circle which touches the axes of coordinates and the line   1 and whose centre lies
3 4
in the first quadrant is x2 + y2  2cx  2cy + c2 = 0 where c is
(A) 1 (B) 2 (C) 3 (D) 6
2 2
563. The points one line x = 2 from which tangents drawn to circle x + y = 16 are at right angles is (are)
(A)  2, 2 7  (B)  2, 2 5 
(C)  2,  2 7  (D)  2,  2 5 
564. The line(s) tangents to the curve y = x2  x
(A) xy=0 (B) x+y=0 (C) xy=1 (D) x+y=1

565. The area of a triangle formed by tangent at any point on the curve and co-ordinate axes is constant, then the
curve may be
(A) a straight line (B) a circle
(C) a parabola (D) a hyperbola

566. The equation of a circle is S1  x2 + y2 = 1. The orthogonal tangents to S1 meet at another circle S2 and
orthogonal tangents to S2 meet at the third circle S3, then
(A) radius of S2 and S3 are in the ratio 1 : 2 (B) radius of S2 and S3 are in the ratio 1 : 2
(C) the circles S1, S2 and S3 are concentric (D) None of these

567. If the area of the quadrilateral formed by the tangents from the origin to the circle x2 + y2 + 6x  8y +  = 0 and
the pair of radii at the point of contact of these tangent to the circle is 2 6 sq. units, then the value of  must
be
(A) 4 6 (B) 24 (C) 1 (D) 12 6
568. The locus of the point of intersection of the tangents at the extremities of the chord of the circle x2 + y2 = a2
which touches the circle x2 + y2  2ax = 0 passes through the point
a   a
(A)  , 0 (B)  0, 
2   2
(C) (0, a) (D) (a, 0)
2 2
569. The equation of the tangent to the hyperbola 3x  y = 3 parallel to the line y = 2x + 4 is
(A) y = 2x + 3 (B) y = 2x + 1
(C) y = 2x  1 (D) y = 2x + 2

x2 y 2
570. From a point P, the chord of contact to the ellipse   a  b …(1)
a b
x2 y2
touches the ellipse 2
  1 …(2) then the locus of the point P is
a b2
(A) director circle of (1) (B) auxillary circle of (2)
(C) x2 + y2 = (a + b)2 (D) x2 + y2 = a2 + b2

571. The point (1,  3) is inside the circle S, S  x2 + y2  8x + 4y + k = 0. What are the possible values of k if the
circle S neither touches the axes nor cuts them?
(A) 5 (B) 6
(C) 7 (D) 8

x2 y2
572. AB and CD are two equal and parallel chords of the ellipse   1. Tangents to the ellipse at A and B
a2 b2
intersect at P and at C and D at Q. The line PQ
(A) passes through the origin (B) is bisected at the origin
(C) cannot pass through the origin (D) is not bisected at the origin
2 2 2
573. If the equation ax – 6xy + y + bx + cy + d = 0 represents pair of lines whose slopes are m and m , then value
of a is / are
(A) a = – 8 (B) a = 8
(C) a = 27 (D) a = – 27

574. The tangent to the hyperbola, x2  3y2 = 3 at the point ( 3, 0) when associated with two asymptotes
constitutes :
(A) isosceles triangle
(B) an equilateral triangle
(C) a triangles whose area is 3 sq. units
(D) a right isosceles triangle.

575. Variable chords of the parabola y2 = 4ax subtend a right angle at the vertex. Then:
(A) locus of the feet of the perpendiculars from the vertex on these chords is a circle
(B) locus of the middle points of the chords is a parabola
(C) variable chords passes through a fixed point on the axis of the parabola
(D) none of these
SECTION - 3: (COMPREHENSION TYPE)

C O M P R E H E N S IO N - 1
Paragraph for Questions Nos. 576 to 578

The straight line(s) which passes through a given point (a, b) and make a given angle  with the given straight line y =
mx + c where m = tan  [(0 <  < 90) and   0, 90], then

576. How many such lines are possible

(A) one (B) two (C) infinite (D) None of these

577. If  is the angle made by the line L with positive direction of x-axis, then tan  is equal to
tan   m tan   m m  tan  m  tan 
(A) (B) (C) (D)
1  m tan  tan   m m  tan  1  m tan 

578. The equation of line L is

m  tan  tan   m
(A) (y  b) = (x  a) (B) (y  b) = (x  a)
1  m tan  tan   m
m  tan  m  tan 
(C) (y  b) = (x  a) (D) (y  b) = (x  a)
m  tan  1  m tan 

C O M P R E H E N S IO N - 2

Paragraph for Questions Nos. 579 to 581

x2 y2
A boy moving along the ellipse   1 at each and every point of it he is drawing a tangent and finding the area
a2 b2
of triangle formed by it with co-ordinate axes at point P, Q, R and S starting from positive axis anticlockwise. He found
that area of triangle is  m and is m at P, Q, R and S.

579. The co-ordinate of point Q are

 a b   a b 
(A)  ,  (B)  , 
 2 2  2 2
 a b   a b 
(C)  ,   (D)  ,  
 2 2  2 2

580. The area of  formed by P, Q and S is

(A) ab (B) ab
(C) ab (D)  ab

581. Equation of normal to the ellipse at S is

b2 a2  b2
(A) ax + by = (B) ax + by =
2 2
a2 b2  a2
(C) bx + ay = (D) bx + ay =
2 2
 C O M P R E H E N S IO N - 3

Paragraph for Questions Nos. 582 to 584

Let an ellipse having major axis and minor axis parallel to x-axis and y-axis respectively. Its two foci S and S are (2,
1), (4, 1) and a line x + y = 9 is a tangent to this ellipse at point P.

582. Eccentricity of the ellipse is

1 1
(A) (B)
12 13
1
(C) (D) None of these
2

583. Length of major axis

(A) 13 (B) 2 11
(C) 52 (D) 2 12

584. The latus rectum of ellipse

12 12
(A) (B)
13 13
24 25
(C) (D)
13 13

C O M P R E H E N S IO N - 4
Paragraph for Questions Nos. 585 to 587

Perpendiculars are drawn from the focus S of the parabola y = ax 2 + bx + c upon the tangents to the parabola at the
 1 9 3 9
points A(– 1, 0) and B(1, 2) meeting them at the point C   ,  and D  ,  respectively.
 4 4 4 4

585. The co-ordinates of the focus are

1 9 1   9
(A)  ,  (B)  , 2 (C)  0,  (D) (– 1, 2)
2 4 2   4

586. The normals at A and B intersect at a point P. The foot of the third normal through the point P is
1 9  1 5 3 5
(A)  ,  (B)  ,  (C) (0, 2) (D)  , 
2 4  2 4 2 4

587. Area of the region bounded by the parabola and the x– axis is
5
(A) (B) 5
4
5 9
(C) (D)
2 2
 C O M P R E H E N S IO N - 5
Paragraph for Questions Nos. 588 to 590

x2 2
Consider an ellipse  y 2   ; ( is parameter > 0) and a parabola y = 8x. If a common tangent to the ellipse and
4
the parabola meets the co-ordinate axes at A and B respectively, then

588. Locus of mid point of AB is

(A) y2 = – 2x
(B) y2 = – x
x
(C) y2 = 
2
x2 y 2
(D)  1
4 2

 2 
589. If the eccentric angle of a point on the ellipse where the common tangent meets it is   , then  is equal to
 3 
(A) 4 (B) 5
(C) 26 (D) 36

590. If two of the three normals drawn from the point (h, 0) on the ellipse to the parabola y2 = 8x are perpendicular,
then
(A) h = 2 (B) h = 3
(C) h = 4 (D) h = 6

C O M P R E H E N S IO N - 6
Paragraph for Questions Nos. 591 to 593

Two straight lines rotate about two fixed points (– a, 0) and (a, 0). If they start from their position of coincidence such
that one rotates at the rate double that of the other, then

591. The point (– a, 0) always lies

(A) inside the curve (B) Outside the curve
(C) on the curve (D) None of these

592. Locus of the curve is

(A) circle (B) straight line
(C) parabola (D) ellipse

593. Distance of the point (a, 0) from the variable point on the curve is
(A) 0 (B) 2a
(C) 3a (D) 4a
 C O M P R E H E N S IO N - 7
Paragraph for Questions Nos. 594 to 596

Consider a point P on a parabola such that 2 of the normals drawn from it to the parabola are at right angles on
parabola, then
2
594. If the equation of parabola is y = 8x, then locus of P is
(A) x2 = 4 (y  6) (B) y2 = 2 (x  6)
(C) y2 = 8 (x  6) (D) 2x2 = (y  6)

595. The ratio of latus rectum of given parabola and that of made by locus of point P is
(A) 4 : 1 (B) 2 : 1
(C) 16 : 1 (D) 1 : 1

596. If P  (x1, y1), the slope of third normal is

y1 y1
(A) (B)
8 2
y1 y1
(C)  (D) 
8 2

C O M P R E H E N S IO N - 8
Paragraph for Questions Nos. 597 to 599

Read the following writeup carefully:

Observe the following facts for a parabola.

(i) Axis of the parabola is the only line which can be the perpendicular bisector of the two chords of the parabola.
(ii) If AB and CD are two parallel chords of the parabola and the normals at A and B intersect at P and the
normals at C and D intersect at Q, then PQ is a normal to the parabola.

597. The vertex of the parabola passing through (0, 1), (– 1, 3), (3, 3) and (2, 1) is
 1 1 
(A)  1,  (B)  , 1
 3 3 
(C) (1, 3) (D) (3, 1)

598. The directrix of the parabola is

1 1
(A) y– =0 (B) y+ =0
24 24
1 1
(C) y+ =0 (D) y– =0
12 12

599. For the parabola y2 = 4x, AB and CD are any two parallel chords having slope 1. C1 is a circle passing
through O, A and B and C2 is a circle passing through O, C and D. C1 and C2 intersect at
(A) (4, – 4) (B) (– 4, 4)
(C) (4, 4) (D) (– 4, – 4)
 C O M P R E H E N S IO N - 9
Paragraph for Questions Nos. 600 to 602

To the circle x2 + y2 = 4 two tangents are drawn from P(4, 0), which touches the circle at T1 and T2 and a rhombus
PT1 PT2 is completed.

600. Circum centre of the triangle PT1T2 is at

(A) (2, 0) (B) (2, 0)
 3 
(C)  , 0  (D) None of these
 2 

601. Ratio of the area of triangle PT1P to that the PT1T2 is

(A) 2:1 (B) 1:2
(C) 3 :2 (D) None of these

602. If P is taken to be at (h, 0) such that P lies on the circle, the area of the rhombus is
(A) 6 3 (B) 2 3
(C) 3 3 (D) None of these

C O M P R E H E N S IO N - 1 0
Paragraph for Questions Nos. 603 to 605

A circle C whose radius is 1 unit, touches the xaxis at point A. The centre Q of C lies in first quadrant. The tangent
from origin O to the circle touches it at T and a point P lies on it such that OAP is a right angled triangle at A and its
perimeter is 8 units.

603. The length of QP is

1 4
(A) (B)
2 3
5
(C) (D) None of these.
3

604. Equation of circle C is

(A) {x  (2  3 )} 2  (y  1)2  1 (B) {x  ( 3  2 )} 2  (y  1)2  1

(C) (x  3 )2  (y  2)2  1 (D) None of these.

605. Equation of tangent OT is

(A) x  3y  0 (B) x  2y  0

(C) y  3x = 0 (D) None of these

 C O M P R E H E N S IO N - 1 1
Paragraph for Questions Nos. 606 to 608

If a circle with centre C(, ) intersects a rectangular hyperbola with centre L (h, k) at four points
P(x1, y1), Q(x2, y2), R(x3, y3) and S(x4, y4), then the mean of the four points P, Q, R, S is the mean of the points C and
L. In other words, the mid-point of CL coincides with the mean point of P, Q, R, S. Analytically
x1  x 2  x 3  x 4   h y1  y 2  y 3  y 4   k
 ;  .
4 2 4 2

606. Five points are selected on a circle of radius a. The centres of the rectangular hyperbolas, each passing
through four of these points lie on a circle of radius
a a
(A) a (B) 2a (C) (D)
2 2

607. A, B, C and D are the points of intersection of a circle and a rectangular hyperbola which have different
centres. If AB passes through the centre of the hyperbola, then CD passes through
(A) centre of the hyperbola (B) centre of the circle
(C) mid-point of the centres of the circle and hyperbola
(D) none of these
2 2 2
608. A circle with fixed centre (3h, 3k) and of variable radius cuts the rectangular hyperbola x  y = 9a at the
points A, B, C , D. The locus of the centroid of the triangle ABC is given by
2 2 2 2 2 2
(A) (x  2h)  (y  2k) = a (B) (x  h)  (y  k) = a
x2 y2 x2 y2
(C)   a2 (D)   a2
h2 k2 h2 k2

C O M P R E H E N S IO N - 1 2

Paragraph for Questions Nos. 609 to 611

The vertices of a ABC lies on a rectangular hyperbola such that the orthocentre of the triangle is (3, 2) and the
asymptotes of the rectangular hyperbola are parallel to the coordinate axis. If the two perpendicular tangents of the
hyperbola intersect at the point (1, 1).

609. The equation of the asymptotes is

(A) xy  1 = x  y (B) xy + 1 = x + y
(C) 2xy = x + y (D) None of these

610. Equation of the rectangular hyperbola is

(A) xy = 2x + y  2
(B) 2xy = x + 2y + 5
(C) xy = x + y + 1
(D) None of these

611. Number of real normals that can drawn from the point (1, 1) to the rectangular hyperbola is
(A) 4 (B) 0
(C) 3 (D) 2
 C O M P R E H E N S IO N - 1 3
Paragraph for Questions Nos. 612 to 614

Let ABCD be a parallelogram whose diagonals equations are AC: x + 2y = 3; BD: 2x + y = 3. If length of diagonal AC
= 4 units and area of ABCD = 8 sq. units.

612. The length of other diagonal BD is

10 20
(A) (B) 2 (C) (D) None of these
3 3

613. The length of side AB is equal to

2 58 4 58 3 58 4 58
(A) (B) (C) (D)
3 9 9 9

614. The length of BC is equal to

2 10 4 10
(A) (B)
3 3
8 10
(C) (D) None of these
3
C O M P R E H E N S IO N - 1 4
Paragraph for Questions Nos. 615 to 617

A coplanar beam of light emerging from a point source have equation x – y + 2(1 + ) = 0,   R. the rays of the
beam strike an elliptical surface and get reflected. The reflected rays form another convergent beam having equation
x – y + 2(1 – ) = 0,   R. Further it is found that the foot of the perpendicular from the point (2, 2) upon any tangent
2 2
to the ellipse lies on the circle x + y – 4y – 5 = 0.

615. The eccentricity of the ellipse is equal to

1 1
(A) (B)
3 3
2 1
(C) (D)
3 2

616. The area of the largest triangle that an incident ray and the corresponding reflected ray can enclose with the
axis of the ellipse is equal to
(A) 4 5 (B) 2 5
(C) 5 (D) None of these

617. Total distance travelled by an incident ray and the corresponding reflected ray is the least if the point of
incidence coincides with
(A) an end of the minor axis (B) an end of the major axis
(C) an end of the latus rectum (D) None of these
 C O M P R E H E N S IO N - 1 5
Paragraph for Questions Nos. 618 to 620

 
A circular arc of radius 2 units subtend an angle of x radians at the centre O such that x   0,  . Tangents at the
 2
end points P and Q of the arc intersect at R. Let f(x) be the area of triangle PQR and let (x) be the area of region
enclosed by the chord PQ and the arc PQ.

618. Length of OR is given by

x x x x
(A) tan (B) 2 tan (C) 2 sec (D) 2 cot
2 2 2 2

619. (x) will be given by

x x x x
(A) x – sinx (B) 2(x – sinx) (C)  tan (D) tan sec
2 2 2 2

 x
620. lim is equal to
x 0 f x
4 3 2 3
(A) (B) (C) (D)
3 4 3 2

SECTION - 4 (MATRIX MATCH Type)

621. Match the following:
List – I List – II
2
(A) The length of latus rectum of parabola 2y + 3y  4x  3 = 0 is (i) 3/4
(x  3)2 (y  4)2 (ii) 2
(B) The length of tangent from point (0,  1) to the circle   1 is
52 52
(C) The length of shortest focal chord of parabola y2 = 4x  3 is (iii) 3
(D) Straight line with slope m represents the locus of middle points of chords of (iv) 4
2 2
hyperbola 3x  2y + 4x  6y = 0 parallel to y = 2x, then m is

622. Match the following:

List – I List – II
2
(x  1) (y  2) 2 (i) 0
(A) The length of common chord of the ellipse   1 and circle
9 4
(x  1)2 + (y  2)2 = 1 is
(B) cosec2 A  cot2 A  sec2 A  tan2 A  (cot2 A  tan2 A) (sec2 A cosec2 A  1) (ii) 1
is
p b c (iii) 2
(C) If a  p, b  q, c  r a q c  0 ,
a b r

p q r
then   is equal to
p  a q  b r 1
(D) The circle x2 + y2 = 4x  7y + 12 = 0 cuts an intercept on y-axis of length (iv) 3
623. Match the following:
List – I List – I
(A) The value of ‘a’ for which the image of point (a, a, 1) w.r.t. the (i) 3
2
line mirror 3x + y = 6a is is the point (a + 1, a) is
(B) The normal chord at a point ‘t’ on the parabola 16y2 = x subtends (ii) 2
right angle at the vertex. Then t2 is equal to
(C) If e and e be the eccentricity of a hyperbola and its conjugate. (iii) 5
1 1
Then 2
 
e e2
(D) The shortest distance between parabola y2 = 4x and circle x2 + y2 + (iv) 1
6x  12y + 20 = 0 is 4 2  A. The A is

624. Match the following:

List – I List – II
(A) Let  = lim lim cos2m n x , where x is rational,  = lim lim cos2m n x , (i) 56
m  n m  n

where x is irrational, then the area of the triangle having vertices (, ), ( – 2, 1)
and (2, 1) is
(B) If the circumference of the circle x2 + y2 + 8x + 8y – b = 0 is bisected by the circle (ii) 2
2 2
x + y – 2x + 4y + a = 0, then |a + b| =
(C) Given a circle of radius 3, tangents are drawn from points A and B lying on one of (iii) 90
its diameters which meet at a point P lying on another diameter perpendicular to the
other diameter. The minimum area of triangle PAB is
(D) If the radius of the circle (x – 1)2 + (y – 2)2 = 1 and (x – 7)2 + (y – 10)2 = 4 are (iv) 18
increasing uniformly w.r.t. time as 0.3 and 0.4 unit/sec, if they touch each other
internally after t sec. then t is equal to

625. Match the following:

ABC be a triangle with a = 3, b = 4, c = 5
List I ListII
(A) Distance between circumcentre and orthocentre (i) 1/3
(B) Distance between centroid and circumcentre (ii) 5/2
(C) Distance between centroid and incentre (iii) 5/6
(D) Distance between centroid and orthocentre (iv) 5/3

626. Match the following curve with their corresponding orthogonal trajectory

List – I List – II
2 3
(A) ay = x (i) y = kx
2
(B) y = ax (ii) y2/3 – x2/3 = c
(C) x2/3 + y2/3 = a2/3 (iii) y2/3 + x2/3 = c
2 2 2
(D) x + y = a (iv) 2y2 + x2 = c
(v) 3y2 + 2x2 = c
627. Match the following
List – I List – II
(A) For a rectangular hyperbola xy = c2 (c is purely imaginary) if a point on it (i) 3
is (a, ) where ‘a’ is a positive number,  can take value
(B) If sides of a triangle are in A.P. and (a – b + c)s = kb2 (where s is the semi 1 3
(ii)
perimeter) then k is equal to 2
(C) Radius of the circle having centre (3, 4) and touching the circle x2 + y2 = 4 (iii) 17
can be
2 2 3
(D) Maximum distance of any point on the circle  x  7    y  2 30   16 (iv)
2
from the centre of the ellipse 25x2 + 16y2 = 400 is

628. Match the following

x2 y 2 (1) 
(A) The number of rational points on the ellipse   1 is
9 4
x2 y 2 (2) 4
(B) Number of integral points on the ellipse   1 is
9 4
x2 (3) 0
(C) Number of rational points on the curve  y2  1 is
3
x2 (4) 2
(D) Number of integral points on the curve  y2  1 is
3

629. Match the following:

List  I List  II
2 2
(A) The triangle PQR is inscribed in the circle x + y = 169. If Q and R have 
(i)
coordinates (5, 12) and (12, 5) respectively find QPR 4
(B) What is the angle between the line joining origin to the point of intersection of 
(ii)
the line 4x + 3y = 24 with circle (x  3)2 + (y  4)2 = 25 2
(C) Two parallel tangents drawn to given circle are cut by a third tangent. The (iii) 
angle subtended by the third tangent at the centre is
(D) For a circle if a chord is drawn along the point of contact of tangents drawn 
(iv)
 6
from a point P. If the chord subtends an angel then find the angle at P
2

630. Match the following:

List – I List – II
x2 y2 (i)
1
(A) The normal at an end of a latusrectum of the ellipse   1 passes
a2
b 2 9
4
through an end of the minor axis if e is equal to
(B) PQ is a double ordinate of a parabola y2 = 4ax. If the locus of its points of 1
(ii)
trisection is another parabola length of whose latus rectum is k times the a
length of the latus rectum of the given parabola then k is equal to
(C) If e and e are the distances of the extremities of any focal chord from the (iii) 1
1 1
focus f of the parabola y2 = 4ax, then  is equal to
e e
 (D) If e and e be the eccentricities of a hyperbola and its conjugate, then (iv) 1  e2
1 1
2
 2 is equal to
e e

631. Match the following:

List – I List – II
2 2
x y (i) 2
(A) The equation of tangent to the ellipse   1 which cuts off
25 16
equal intercepts on axes is x – y = a where a equal to
(B) The normal y = mx – 2am – am3 to the parabola y2 = 4ax subtends a (ii) 3
right angle at the vertex if m equals to
(C) The equation of the common tangent to parabola y2 = 4x and x2 = 4y (iii) 8
k
is x + y + = 0, then k is equal to
3
(D) An equation of common tangent to parabola y2 =8x and the (iv) 41
k
hyperbola 3x2 – y2 = 3 is 4x – 2y + = 0, then k is equal to
2

632. The parabola y2 = 4ax has a chord AB joining points A at12 , 2at1    
and B at 22 , 2at 2 . Then match the

following
List – I List – II
1
(A) AB is a normal chord (i) t2 =  t1 
2
4
(B) AB is a focal chord (ii) t2 = 
t1
1
(C) AB subtends 90 at point (0, 0) (iii) t2 = 
t1
2
(D) AB is inclined at 45 to the axis of parabola (iv) t2 =  t1 
t1

Section-5 : (INTEGER type)

AP
633. A point P moves in such a way that  3 where A(1, 2) and B(3, 5), then maximum distance of P from A is
PB
__________

634. The line L has intercepts 1 and 1/2 on the co-ordinate axes. When keeping the origin fixed, the co-ordinate
axes are rotated through a fixed angle, if the same line has intercepts p and q on the rotates axes. Then
1 1
2
 is _____________
p q2

635. If PSQ is the focal chord of the parabola y2 = 8x such that SP = 6, then the length SQ is ______________
636. If the ellipse x2 + 81y2  81 = 0 and the circle x 2 + y2  9 = 0 intersect an angle  in first quadrant, then the
value of ( 3 tan ) is ____________
2 2 2 2
637. Let (xi, yi) where i = 1, 2, 3, 4 are the integral solutions of equation 2x y + y – 6x – 12 = 0. The area of
quadrilateral whose vertices are (xi, yi), i = 1, 2, 3, 4 is _______

638. Square of diameter of the circle having tangent at (1, 1) as x + y  2 = 0 and passing through (2, 2) is
________


639. If P be a point on ellipse 4x2 + y2 = 8 with eccentric angle . Tangent and normal at P intersects the axes at
4
A, B, A and B respectively, then the ratio of area of APA and area of BPB is ___________ .

x2 y2
640. P is the positive extremity of the latus rectum of the ellipse   1 and A is the positive major vertex and
25 16
B is the positive minor vertex then the 10 times of the area bounded by BPA and chords BP and AP is
___________

641. The minimum of the distances from the point (0, 1) to the points of intersection of the lines (3cos + 4sin) x +
(2cos + 2sin) y  (5cos + 6sin) = 0, where different values of  gives different lines, is ____________

642. A line passes through the point P (2, 3) and makes an angle  with positive direction of x-axis. If it meets the
lines represented by x2  2xy  y2 = 0 at the points A and B. If PA  PB = 17, then the value of  in degrees is
________________

6 6
643. Six points (xi, yi), i = 1, 2, …, 6 are taken on the circle x 2 + y2 = 4 such that  xi  8 and y i  4 . The line
i1 i1

segment joining orthocentre of a triangle made by any three points and the centroid of the triangle made by
other three points passes through a fixed points (h, k), then h + k is ________

644. The square of the length of the intercept on the normal at the point P(18, 12) of the parabola y2 = 8x made by
the circle on the line joining the focus and P as diameter, is ___________


645. The angle between the straight lines x cos + y sin = p and ax + by + p = 0 is . They meet the straight line
4
x sin – y cos = 0 in the same point, then the value of a2 + b2 is ___________

646. Area of the rectangle formed by a asymptotes of the hyperbola xy  3y  2x = 0 and coordinate axes is

647. The tangent at the point A(12, 6) to a parabola intersects its directrix at the point B(– 1, 2). The focus of the parabola
lies on x-axis. The number of such parabolas is

648. The circle x2 + y2 = 4 cuts the circle x2 + y2  2x  4 = 0 at the point A and B. If the circle x2 + y2  4x  k = 0
passes through A and B, then the value of k is

AB 2  BC2  CA 2
649. If G is the centroid of ABC with vertices A(a, 0), B( a, 0) and C(b, c) then =
GA 2  GB2  GC 2

650. A man running round a race course notes that the sum of the distance of two flag posts from him is always 10 meter
and distance between flag posts is 8 m. The area of the path, he encloses (in square meters) is k. What is the value of
k?
 Answer Key

Qs. Ans. Qs. Ans. Qs. Ans.

530 200 551 C 601
531 B 552 A 602 A
532 B 553 D 603 C
533 C 554 D 604 A
534 D 555 B 605 C
535 C 556 AD 606 D
536 A 557 AB 607 B
537 D 558 AC 608 C
538 B 559 D 609 B
539 D 560 AC 610 C
540 A 561 AB 611 D
541 C 562 AD 612 C
542 A 563 AC 613 A
543 B 564 BC 614 A
544 D 565 AD 615 C
545 C 566 AC 616 B
546 A 567 BC 617 D
547 B 568 AC 618 C
548 A 569 BC 619 B
549 C 570 AC 620 C
550 B 571 ABCD 621 A-(ii), B-(iii), C-(iv), D-(i)
572 AB 622 A-(i), B-(i), C-(iii), D-(ii)
573 BD 623 A-(ii), B-(ii), C-(iv), D-(iii)
574 BC 624 A-(ii), B-(i), C-(iv), D-(iii)
575 ABC 625 A-(ii), B-(i), C-(iv), D-(iv)
576 B 626 A-(), B-(), C-(), D-()
577 A 627
578 D 628 A-(1), B-(2), C-(1), D-(3)
579 B 629 A-(i), B-(ii), C-(ii), D-(ii)
580 A 630 A-(iv), B-(i), C-(ii), D-(iii)
581 B 631
582 B 632 A-(iv), B-(iii), C-(ii), D-(i)
583 C 633
584 C 634 5
585 B 635 3
586 D 636 8
587 D 637 16
588 638 2
589 639 4
590 640 17
591 C 641 1
592 A 642 90
593 B 643 3
594 B 644 40
595 A 645 2
596 B 646 6
597 A 647 4
598 B 648 4
599 A 649 3
600 A 650 15