MATHEMATICS

RPP
REVISION PRACTICE PROBLEMS
Crash Course JEE/CET 2025
Topic: Parabola 8.1
Type-1 Basics Of Conic Section AND Parabola
1. locus of the mid-point of the line segment joining the focus of the parabola y2 = 4ax to a
moving point of the parabola, is another parabola whose direcrix is : [JEE- Main 2021]
a a
(1) x  – (2) x  (3) x = 0 (4) x = a
2 2

2. If P is a point on the parabola y = x2 + 4 which is closest to the straight line y = 4x – 1, then

the co-ordinates of P are :
(1) (3, 13) (2) (1, 5) (3) (–2, 8) (4) (2, 8)

3. The equation of the circle drawn with the focus of the parabola (x  1)2  8 y = 0 as its centre
and touching the parabola at its vertex is
(1) x2 + y2  4 y = 0 (2) x2 + y2  4 y + 1 = 0
2 2
(3) x + y  2 x  4 y = 0 (4) x2 + y2  2 x  4 y + 1 = 0

4. Which one of the following equations represented parametrically, represents equation to a

parabolic profile?
t
(1) x = 3 cos t ; y = 4 sin t (2) x2  2 =  2 cos t ; y = 4 cos2
2
x = tan t ; t t
(3) y = sec t (4) x = 1  sin t ; y = sin + cos
2 2
5. Let A (4,–4) and B(9,6) be points on the parabola, y2 = 4x. Let C be chosen on the arc AOB of
the parabola, where O is the origin, such that the area of ACB is maximum. Then, the area
(in sq. units) of ACB, is
1 3 1
(A) 31 (2) 32 (3) 31 (4) 30
4 4 2
6. Axis of a perpendicular lies along X-axis. It its vertex and focus are at distance 2 and 4
respectively from the origin, on the positive X-axis, then which of the following points does not
lie on it ?

(1) (4,–4) (2) (6,4 2 ) (3) (8,6) (4) (5,2 6)

7. The maximum area (in sq. units) of a rectangle having its base on the X-axis and its other two
vertices on the parabola, y = 12 – x2 such that the rectangle lies inside the parabola, is
(1) 36 (2) 20 2 (3) 32 (4) 18 3
 Type-2 Focal Chord
8. If one end of a focal chord of the parabola, y2 = 16x is at (1,4), then the length of this focal
chord is
(1) 22 (2) 25 (3) 24 (4) 20

9. The length of the chord of the parabola x2 = 4y having equation x – 2 y + 4 2 = 0 is

(1) 8 2 (2) 2 11 (3) 3 2 (4) 6 3

10. The length of a focal chord of the parabola y2 = 4ax at a distance b from the vertex is c, then
(1) 2a2 = bc (2) a3 = b2c (3) ac = b2 (4) b2c = 4a3

11. Locus of the feet of the perpendiculars drawn from vertex of the parabola y2 = 4ax upon all
such chords of the parabola which subtend a right angle at the vertex is
(1) x2 + y2 – 4ax = 0 (2) x2 + y2 – 2ax = 0 (3) x2 + y2 + 2ax = 0 (4) x2 + y2 + 4ax = 0

12. The triangle PQR of area 'A' is inscribed in the parabola y2 = 4ax such that the vertex P lies at
the vertex of the parabola and the base QR is a focal chord. The modulus of the difference of
the ordinates of the points Q and R is
A A 2A 4A
(1) (2) (3) (4)
2a a a a

13. If the locus of the middle points of all such chords of the parabola y2 = 8x which has a length
of 2 units, has the equation y4 + ly2(2 – x) = mx – n where l, m, n  N, find the value of
(l + m + n)
1 
14. If one end of a focal chord AB of the parabola y2 = 8x is at A  , –2  , then the equation of
2 
the tangent to it at B is
(1) x – 2y + 8 = 0 (2) x + 2y + 8 = 0 (3) 2x + y – 24 = 0 (4) 2x – y – 24 = 0

15. PQ is a chord of parabola x2 = 4y which subtends right angle at vertex. Then locus of
centroid of triangle PSQ, where S is the focus of given parabola, is
4 4 4
(1) x2 = 4(y + 3) (2) x2 = (y – 3) (3) x2 = (y + 3) (4) x2 = (y + 3)
3 3 3
Type-3 Tangent Of Parabola AND Properties Of Tangents & Director circle
16. A tangent is drawn to the parabola y2 = 6x which is perpendicular to the line 2x + y = 1. Which
of the following points does NOT lie on it ?
(1) (–6,0) (2) (4,5) (3) (5,4) (4) (0,3)

17. If y = mx + 4 is a tangent to both the parabolas, y2 = 4x and x2 = 2by, then b is equal to

(1) –32 (2) –128 (3) –64 (4) 128

18. The tangents to the curve y = (x – 2)2 – 1 at its points of intersection with the line x – y = 3,
intersect at the point
5   5  5   5 
(1)  ,1 (2)  – ,–1 (3)  , –1 (4)  – ,1
2   2  2   2 
19. The straight line joining any point P on the parabola y2 = 4ax to the vertex and perpendicular
from the focus to the tangent at P, intersect at R, then the equation of the locus of R is
(1) x2 + 2y2 – ax = 0 (2) 2x2 + y2 – 2ax = 0
2 2
(3) 2x + 2y – ay = 0 (4) 2x2 + y2 – 2ay = 0

20. The area (in sq. units) of the smaller of the two circles that touches the parabola,
y2 = 4x at the point (1,2) and the X-axis is
(1) 8(3 – 2 2 ) 
(2) 4 3  2  
(3) 8 2 – 2  
(4) 4 2 – 2 
21. Equation of a common tangent to the circle, x2 + y2 – 6x = 0 and the parabola, y2 = 4x, is
(1) 3y  3x  1 (2) 2 3y  12x  1 (3) 3y  x  3 (4) 2 3y  –x – 12

22. If the common tangent of the parabolas, y2 = 4x and x2 = 4y also touches the circle,
x2 + y2 = c2, then c is equal to
1 1 1 1
(1) (2) (3) (4)
2 4 2 2 2

23. Let L1 be a tangent to the parabola y2 = 4(x + 1) and L2 be a tangent to the parabola
y2 = 8(x + 2) such that, L1 and L2 intersect at right angles. Then, L1 and L2 meet on the
straight line
(1) x + 3 = 0 (2) 2x + 1 = 0 (3) x + 2 = 0 (4) x + 2y = 0

24. The equation of director circle of the parabola y2 = 10(x – 1)

(1) x2 + y2 –2x –4 y + 4 = 0 (2) 2x + 3 = 0
(3) 2x + 5 = 0 (4) x2 + y2 – 4x + 2y + 2 = 0

Type-4 Normal To Parabola & Co-Normal Points

25. Tangent and normal are drawn at P(16, 16) on the parabola y2 = 16x, which intersect the axis
of the parabola at A and B, respectively. If C is the centre of the circle through the points P,
A and B and ∫CPB = ,θthen a value of tan θ is:
4 1
(1) 3 (2) (3) (4) 2
3 2

26. If the parabolas y2 = 4b(x – c) and y2 = 8ax have a common normal, then which one of the
following is valid choice for the ordered triad (a,b,c)?
1  1 
(1)  ,2,0  (2) (1,10) (3) (1,1,3) (4)  ,2,3 
2  2 
27. If the normal to a parabola y2 = 4ax at P meets the curve again in Q and if PQ and the normal
at Q makes angles  and  respectively with the x-axis then tan (tan  + tan ) has the
value equal to
1
(1) 0 (2) – 2 (3) – (4) – 1
2

28. Equation of the other normal to the parabola y2 = 4x which passes through the intersection of
those at (4,  4) and (9,  6) is
(1) 5x  y + 115 = 0 (2) 5x + y  135 = 0
(3) 5x  y  115 = 0 (4) 5x + y + 115 = 0
29. The normal chord of a parabola y2 = 4ax at the point whose ordinate is equal to the abscissa,
then angle subtended by normal chord at the focus is :
 
(1) (2) tan 1 2 (3) tan 1 2 (4)
4 2

30. If two normals to a parabola y2 = 4ax intersect at right angles then the chord joining their feet
passes through a fixed point whose co-ordinates are :
(1) ( 2a, 0) (2) (a, 0) (3) (2a, 0) (4) none

31. Let P be a point on the parabola, y2 = 12x and N be the foot of the perpendicular drawn from
P on the axis of the parabola. A line is now drawn through the mid-point M and PN, parallel to
4
its axis which meets the parabola at Q. If the y-intercept of the line NQ is , then
3
1 1
(1) PN = 4 (2) MQ  (3) MQ  (4) PN  3
3 4

32. If the three normals drawn to the parabola, y2 = 2x pass through the point (a,0) a  0, the 'a'
must be greater than :
1 1
(1) (2)  (3) –1 (4) 1
2 2
Type-5 Pair Of Tangent, Chord Of Contact, Diameter
33. From an external point P, pair of tangent lines are drawn to the parabola, y2 = 4x. If 1 & 2 are

the inclinations of these tangents with the axis of x such that, 1 + 2 = , then the locus of P
4
is :
(1) x  y + 1 = 0 (2) x + y  1 = 0 (3) x  y  1 = 0 (4) x + y + 1 = 0

34. The points of contact Q and R of tangent from the point P (2, 3) on the parabola y2 = 4x are
1
(1) (9, 6) and (1, 2) (2) (1, 2) and (4, 4) (3) (4, 4) and (9, 6) (4) (9, 6) and ( , 1)
4

35. Tangents are drawn from the point ( 1, 2) on the parabola y2 = 4 x. The length , these
tangents will intercept on the line x = 2 is :
(1) 6 (2) 6 2 (3) 2 6 (4) none of these
 MATHEMATICS

RPP
Crash Course JEE/CET 2025 REVISION PRACTICE PROBLEMS

Topic: Ellipse 8.2

Type-1 Basic of Ellipse & Equation & Eccentric angle, auxilliary circle,
corresponding point

1. If the coordinates of two points A and B are ( 7 , 0) and (– 7 ,0) respectively and P is any
point on the conic, 9x2 + 16y2 = 144, then PA + PB is equal to
(1) 16 (2) 8 (3) 6 (4) 9

x 2 y2
2. Let  = 1 (a > b) be a given ellipse, length of whose latus rectum is 10. If its
a2 b2
5
eccentricity is the maximum value of the function,   t    t – t 2 , then a2 + b2 is equal to
12
(1) 145 (2) 116 (3) 126 (4) 135

4
3. The length of the mirror axis (along y – axis) of an ellipse in the standard from is . If this
3
ellipse touches the line, x + 6y = 8 ; then its eccentricity is
5 1 11 1 11 1 5
(1) (2) (3) (4)
6 2 3 3 3 2 3
y2
4. The eccentricity of the ellipse (x – 3)2 + (y – 4)2 = is
9
3 1 1 1
(1) (2) (3) (4)
2 3 3 2 3

x2 y 2
5. F1 and F2 are the two foci of the ellipse   1. Let P be a point on the ellipse such that
9 4
PF1  2 PF2 , where F1 and F2 are the two foci of the ellipses. The area of PF1F2 is

5 13
(1) 3 (2) 4 (3) (4)
2

6. Let S(5, 12) and S'(– 12, 5) are the foci of an ellipse passing through the origin.
The eccentricity of ellipse equals
1 1 1 2
(1) (2) (3) (4)
2 3 2 3
7. The y-axis is the directrix of the ellipse with eccentricity e = 1/2 and the corresponding focus
is at (3, 0), equation to its auxilary circle is
(1) x2 + y2 – 8x + 12 = 0 (2) x2 + y2 – 8x – 12 = 0
2 2
(3) x + y – 8x + 9 = 0 (4) x2 + y2 = 4

8. An ellipse with foci at (0,2) and (0,–2) and minor axis of length 4, passes through which of the
following points ?
(1)  2,2  
(2) 2, 2  
(3) 2,2 2  
(4) 1,2 2 
9. Let S and S’ be the foci of an ellipse and B be any one of the extremities of its minor axis. If
S’ BS is a right angled triangle with right angle at B and area (’S BS) = 8 sq units, then the
length of a latus rectum of the ellipse is
(1) 2 2 (2) 4 2 (3) 2 (4) 4

10. Let the length of the latus rectum of an ellipse with its major axis along X-axis and centre at
the origin, be 8. If the distance between the foci of this ellipse is equal to the length of its
minor axis, then which one of the following points lies on it ?

(1) 4 2,2 3  
(2) 4 3,2 2  
(3) 4 2,2 2  
(4) 4 3,2 3 
Type-2 Tangent of ellipse & property of tangents & director circle

11. Let L be tangent line to the parabola y2 = 4x – 20 at (6, 2). If L is also a tangent to the ellipse
x2 y2
  1, then the value of b is equal to : [JEE- Main 2021]
2 b
(1)11 (2) 14 (3) 16 (4) 20

x2 y2  –9 
12. If the line x – 2y =12 is tangent to the ellipse   1 at the point  3,  , then the length
a 2 b2  2 
of the latusrectum of the ellipse is

(1) 8 3 (2) 9 (3) 5 (4) 12 2

13. If the tangent on the ellipse 4x2 + y2 = 8 at the points (1,2) and (a,b) are perpendicular to each
other, then a2 equal to
128 64 4 2
(1) (2) (3) (4)
17 17 17 17
x 2 y2
14. Tangents are drawn to the ellipse 
16 12
 
 1 from the point R 8, 2 7 . Find the distance

which these tangents, intercept on the ordinate through nearer focus of the ellipse.

15. Given ellipse x2 + 4y2 = 16 and parabola y2 – 4x – 4 = 0

The quadratic equation whose roots are the slopes of the common tangents to parabola and
ellipse, is
(1) 3x2 – 1 = 0 (2) 5x2 – 1 = 0
2
(3) 15x + 2x – 1 = 0 (4) 2x2 – 1 = 0
 x2 y2
16. The Locus of the middle point of chords of an ellipse   1 passing through P(0, 5)
16 25
is another ellipse E. The coordinates of the foci of the ellipse E, is
3 3
(1)  0,  and  0,  (2) (0, – 4) and (0, 1)
 5   5 
11 1
(3) (0, 4) and (0, 1) (4)  0,  and  0, 
 2  2  

x
17. If the circle x2 + y2 – 2x – 4y + k = 0 and director circle of ellipse + y2 = 1 intersects
4
orthogonally then k equals
(1) 0 (2) 5 (3) – 5 (4) 2

x 2 y2
18. If 3x + 4y = 12 2 is a tangent to the ellipse 2  = 1 for some R, then the distance
a 9
between the foci of the ellipse is

(1) 2 7 (2) 4 (3) 2 2 (4) 2 5

19. Let L be a common tangent line to the curves 4x2 + 9y2 = 36 and (2x)2 + (2y)2 = 31. Then the
square of the slope of the line L is ______.

20. If tangents are drawn to the ellipse x2 + 2y2 = 2 at all points on the ellipse other than its four
vertices, then the mid-points of the tangents intercepted between the coordinate axes lie on
the curve
x2 y2 1 1 x2 y2 1 1
(1)  1 (2) 2
 2 1 (3)  1 (4) 2
 2 1
4 2 4x 2y 2 4 2x 4y

21. Which of the following points lies on the locus of the foot of perpendicular drawn upon any
x 2 y2
tangent to the ellipse,  = 1 from any of its foci?
4 2
(1) (–2, 3) 
(2) –1, 2  (3)  –1, 3  (4) (1,2)

Type-3 Normal to ellipse & its property,Important highlight and reflection property
& Micellaneous
22. If the normal at an end of a latus rectum of an ellipse passes through an extremity of the
minor axis, then the eccentricity e of the ellipse satisfies.
(1) e4 + 2e2 –1 = 0 (2) e3 + 2e2 –1 = 0 (3) e4 + e2 –1 = 0 (4) e2 + 2e –1 = 0

1
23. Let x = 4 be a directerix to an ellipse whose centre is at origin and its eccentricity is .
2
If P(1,)  > 0 is a point on this ellipse, then the equation of the normal to it at P is
(1) 8x – 2y = 5 (2) 4x – 3y = 2 (3) 7x – 4y = 1 (4) 4x – 2y = 1
24. Let the line y = mx and the ellipse 2x2 + y2 = 1 intersect at a point P in the first quadrant. If the
 1 
normal to this ellipse at P meets the co-ordinate axes at  – ,0  and (0,), then  is equal
 3 2 
to
2 2 2 2 2
(1) (2) (3) (4)
3 3 3 3

x 2 y2
25. If the point of intersections of the ellipse  2  1 and the circle x2 + y2 = 4b, b > 4 lie on the
16 b
curve y2 = 3x2, then b is equal to:
(1) 12 (2) 5 (3) 6 (4) 10

26. If the point P on the curve, 4x2 + 5y2 = 20 is farthest from the point Q(0,–4), then PQ2 is equal
to
(1) 29 (2) 21 (3) 48 (4) 36

27. If the curve x2 + 2y2 = 2 intersects the line x + y = 1 at two points P and Q, then the angle
subtended by the line segment PQ at the origin is :
  1   1   1   1
(1)  tan–1   (2) – tan–1   (3) – tan–1   (4)  tan–1  
2 3 2 3 2 4 2 4

a2 x2 y 2
28. If tan 1. tan 2 =  then the chord joining two points  1 &  2 on the ellipse   1 will
b2 a2 b2
subtend a right angle at :
(1) focus (2) centre
(3) end of the major axis (4) end of the minor axis

x2 y 2
29. If a tangent having slope 2 of the ellipse   1 (a > 0, b > 0) is normal to the circle
a2 b2
x2 + y2 + 4x + 1 = 0, then the maximum value of ab is equal to
(1) 4 (2) 3 (3) 2 (4) 1
 MATHEMATICS

RPP
Crash Course JEE/CET 2025 REVISION PRACTICE PROBLEMS

Topic: Hyperbola 8.3

Type-1 Basic Of Hyperbola AND Auxiliary Circle, Eccentric Angle, Focal Distance &
Conjugate Hyperbola AND Asymptotes Of Hyperbola
1. The locus of the point of intersection of the lines
 3  kx  ky – 4 3  0 and 3x – y – 4  3  k  0 is a conic, whose eccentricity is______.

2. Locus of the centre of the circle which touches the two circles x2 + y2 + 8x – 9 = 0
and x2 + y2 – 8x + 7 = 0 externally, is
x2 y 2 x2 y 2 x2 y2
(1) x2 + y2 = 16 (2)  =1 (3)  =1 (4)  =1
1 15 1 12 1 15

3. If the eccentricity of the hyperbola x2  y2 sec2  = 5 is times the eccentricity of the ellipse
x2 sec2  + y2 = 25, then a value of  is :
(1) /6 (2) /4 (3) /3 (4) /2

4. If 5x + 9 = 0 is the directrix of the hyperbola 16x2 – 9y2 = 144, then its corresponding focus is
 5  5 
(1)  – ,0  (2) (–5,0) (3)  ,0  (4) (5,0)
 3  3 
 x2 y2
5. Let 0 <  < . If the eccentricity of the hyperbola –  1 is greater than 2, then
2 cos 2  sin 2 
the length of its latus rectum lies in the interval
 3 3 
(1)  1,  (2) (3) (3)  ,2 (4) (2,3]
  2 2 

6. The focal length of the hyperbola x2 – 3y2 – 4x – 6y – 11 = 0, is

(1) 4 (2) 6 (3) 8 (4) 10

x2 y2
7. The hyperbola  1 passes through the point of intersection of the lines,
a2 b2
7x + 13y – 87 = 0 and 5x – 8y + 7 = 0 & the latus rectum is 32 2 /5. Find 'a' & 'b'.
8. If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is
13, then the eccentricity of the hyperbola is
13 13 13
(1) (2) 2 (3) (4)
12 8 6
 y2 x2 
9. Let S   x, y   R2 : 2
–  1
 1 r 1– r 
where r  ±1. Then, S represents
2
(1) a hyperbola whose eccentricity is when 0 < r < 1.
1– r
2
(2) a hyperbola whose eccentricity is when 0 < r < 1.
1 r
2
(3) an ellipse whose eccentricity is , when r > 1.
r 1
1
(4) an ellipse whose eccentricity is , when r > 1.
r 1
10. A hyperbola has its centre at the origin, passes through the point (4,2) and has transverse
axis of length 4 along the X-axis. Then the eccentricity of the hyperbola is
2 3 3
(1) 2 (2) (3) (4)
3 2
11. If a directrix of a hyperbola centred at the origin and passing through the point (4, –2 3 ) is
5x = 4 5 and its eccentricity is e, then
(1) 4e4 – 12e2 – 27 = 0 (2) 4e4 – 24e2 + 27 = 0
(3) 4e4 + 8e2 – 35 = 0 (4) 4e4 – 24e2 + 35 = 0

Type-2 Tangent Of Hyperbola AND Chord Of Contact,Director Circle

12. The locus of the midpoints of the chord of the circle, x2 + y2 = 25 which is tangent to the
x 2 y2
hyperbola, –  1 is :
9 16
(1) (x2 + y2)2 – 16x2 + 9y2 = 0 (2) (x2 + y2)2 – 9x2 + 144y2 = 0
(3) (x2 + y2)2 – 9x2 – 16y2 = 0 (4) (x2 + y2)2 – 9x2 + 16y2 = 0

13. If the eccentricity of the standard hyperbola passing through the point (4,6) is 2, then the
equation of the tangent to the hyperbola at (4,6) is
(1) 3x – 2y = 0 (2) x – 2y + 8 = 0 (3) 2x – y – 2 = 0 (4) 2x – 3y + 10 = 0

14. Find the equations of the tangents to the hyperbola x2  9y2 = 9 that are drawn from
(3, 2). Find the area of the triangle that these tangents form with their chord of contact.
x2 y2
15. A tangent to the ellipse   1 with centre C meets its director circle at P and Q. Then the
9 4
product of the slopes of CP and CQ, is
9 4 2 1
(1) (2) (3) (4) –
4 9 9 4
16. Locus of the point of intersection of the tangents at the points with eccentric angles  and
 x2 y2
 on the hyperbola 2  2 = 1 is :
2 a b
(1) x = a (2) y = b (3) x = ab (4) y = ab

17. Let P be the point of intersection of the common tangents to the parabola y2 = 12x and the
hyperbola 8x2 – y2 = 8. If S and S' denotes the foci of the hyperbola where S lie on the positive
X-axis then P divides SS' in a ratio
(1) 13 : 11 (2) 14 : 13 (3) 5 : 4 (4) 2 : 1

x2 y2
18. A line parallel to the straight line 2x – y = 0 is tangent to the hyperbola –  1 at the
4 2
point (x1,y1). Then x12  5y12 is equal to
(1) 10 (2) 5 (3) 6 (4) 8

x2 y 2
19. P is a point on the hyperbola  = 1, N is the foot of the perpendicular from P on the
a2 b2
transverse axis. The tangent to the hyperbola at P meets the transverse axis at T . If O is the
centre of the hyperbola, the OT. ON is equal to
(1) e2 (2) a2 (3) b2 (4) b2/a2

20. Let the major axis of a standard ellipse equals the transverse axis of a standard hyperbola
and their director circles have radius equal to 2R and R respectively. If e1 and e2 are the
eccentricities of the ellipse and hyperbola then the correct relation is
(1) 4e12 – e22 = 6 (2) e12 – 4e22 = 2 (3) 4e22 – e12 = 6 (4) 2e12 – e22 = 4

Type-3 Normal To Hyperbola AND Diameter & Standard Property Of Hyperbola

x2 y2
21. If the line y = mx + 7 3 is normal to the hyperbola –  1, then a value of m is
24 18

3 15 2 5
(1) (2) (3) (4)
5 2 5 2

22. If a hyperbola passes through the point P(10,16) and it has vertices at (± 6,0), then the
equation of the normal to it at P is
(1) 3x + 4y = 94 (2) x + 2y = 42 (3) 2x + 5y =100 (4) x + 3y = 58


23. Let P (a sec , b tan ) and Q (a sec , b tan ), where  +  = , be two points on the
2
x2 y2
  1 hyperbola . If (h, k) is the point of intersection of the normals at P & Q, then find k.
a2 b2
 Type-4 Reflection Property Of Hyperbola AND Rectangular Hyperbola
24. A square ABCD has all its vertices on the curve x2y2 = 1. The midpoints of its sides also lie on
the same curve. Then, the square of area of ABCD is

25. A hyperbola having the transverse axis of length 2 has the same foci as that of the ellipse
2 2
3x  4y  12 , then this hyperbola does not pass through which of the following points?

 1   3   1   3 1 
(1)  ,0  (2)  – ,1 (3)  1,–  (4)  , 
 2   2   2  2 2
  

26. If x + iy=   i where i = 1 and  and  are non zero real parameters then  = constant
and  = constant, represents two systems of rectangular hyperbola which intersect at an
angle of
   
(1) (2) (3) (4)
6 3 4 2

x2 y 2
27. The reflection of hyperbola  = 1 in the line y = x will be a conic. The eccentricity of that
9 4
conic will be
13 13 5 5
(1) (2) (3) (4)
2 3 2 3