Ekalavya - Conic Sections - Questions

SCQ
Question 1: The radius of a circle passing through the focus of parabola x2 = 4 y and
touching it at the point ( 6, 9) is _____.
Options:
(a) 5
(b) 5 10
(c) 5 5
(d) 10
RELATED QUESTION LINK:

https://drive.google.com/file/d/1TY6LSDz5hiSOoVJlCl86EAIpLBfmekCg/view?usp=sharing
Question 2: The eccentricity of the ellipse 3x 2 + 4 y 2 = 12 is changing at a rate of 0.1 per

second. The time at which it coincides with the auxiliary circle is _____.
Options:
(a) 2 seconds
(b) 3 seconds
(c) 5 seconds
(d) 6 seconds

https://drive.google.com/file/d/1lntiHPTiuRviEYG-zlyA7TVKYH5UciCh/view?usp=sharing
Question 3: A series of ellipses are described with given focus and corresponding directrix.
The locus of the extremities of their minor axes is _____.
Options:
(a) pair of straight line
(b) circle
(c) parabola
(d) ellipse

https://drive.google.com/file/d/1s4dJZ_4p-LXcFxQv09_qWk666-9fzNdi/view?usp=sharing
 5 3 
Question 4: Let 5 x − 3 y = 8 2 be the normal at the point P  ,  to the ellipse
 2 2
x2 y 2
+ = 1, a  b . If M, M are the feet of the perpendiculars from the foci S, S
a 2 b2
respectively on the tangent at P, then the point of intersection of SM and SM is _____.
Options:
5 
(a)  , 0 
2 
 5
(b)  0, 
 2
 41 3 
(c)  , 
 10 2 2 2 
 3 41 
(d)  , 
 2 2 10 2 

https://drive.google.com/file/d/14oaRJTGVJU_Br2ytJL3fOL-L9qInxf55/view?usp=sharing
Question 5: The locus of the centre of the rectangle formed by the tangents and the normals
at the ends of the focal chords of parabola y 2 = 4ax is _____.
Options:
(a) 2ax = y 2 – 2a 2
(b) y = 2a ( x – a )
2
(c) y = 4a ( x – a )
2
(d) y = 4a ( x + a )
2

https://drive.google.com/file/d/1MvZtzyrYQeqqDy7Fsf__KCoDqSMpbypC/view?usp=sharing
x2 y 2
Question 6: Let P be a variable point on the ellipse + = 1 with foci F1 and F2 .If A is
10 1
the area of the triangle PF1F2 , then the maximum value of A is _____.
Options:
(a) 3
(b) 4
(c) 5
(d) 6

Question 7: If the rectangular hyperbola ( x –1)( y – 2) = 4 cuts a circle

x2 + y 2 – 7 x – 9 y + c = 0 at the points ( 3, 4) , ( 5, 3) , ( 2, 6 ) and the point P ( u, v ) , then the
value of 9 ( u + v ) is equal to _____.
Options:
(a) –8
(b) –9
(c) 8
(d) 9

https://drive.google.com/file/d/1788z-ECXtKj8KLbckFb7jYxfK2ORLD6z/view?usp=sharing
Question 8: If three parabolas touch all the three lines x = 0 , y = 0 and x + y = 2 , then the
maximum area of the triangle formed by joining their foci is _____.
Options:
(a) 3
(b) 6
3 3
(c)
4
3 3
(d)
2

https://drive.google.com/file/d/1_qVVR5I7aZn4dy74dFCWicv2au1--2qV/view?usp=sharing
Question 9: A rectangular hyperbola has one of its asymptotes as x – y + 2 = 0 . The tangent

at the point P ( 0, 1) is x = 0 . The length of the latus rectum is _____.
Options:
(a) 2
(b) 3
(c) 2
(d) none of these

Question 10: A ( 0, 2) , B and C are three points on the parabola y = x + 4 and such that
2

 CBA = . The range of the ordinate of C is _____.
2
Options:
(a) ( −,0   4,  )
(b) ( −,0)  ( 4, )
(c) 0, 4
(d) ( −,0)  4, )

https://drive.google.com/file/d/164-NjuRKUqCDDowQz4LdKsDQb-UD_NMS/view?usp=sharing
Question 11: Two congruent circles of the largest possible radii having the following
properties.
(i) They intersect each other orthogonally.
(ii) They touch both the curves 4 ( y + 2) = x and 4 ( 2 – y ) = x in the region
2 2
x   −2 2, 2 2  .
Then, the radii of these circles are each equal to _____.
Options:
(a) 2
(b) 3
1
(c)
3
3
(d)
2

https://drive.google.com/file/d/1xBZIl2xTsHBEfCDpL1j82PrPltB5c6_m/view?usp=sharing
Question 12: AB is a chord of a parabola with vertex A. The axis of the parabola is inclined
at angle of 30o with the X axis. BC is drawn perpendicular to AB, meeting the axis of the
parabola at C. If the length of the perpendicular drawn from the focus to the directrix is 8
units, then the projection of BC on the axis of parabola is _____ units long.
Options:
(a) 8
(b) 16
(c) 24
(d) 32

Question 13: The locus of the focus of an ellipse with length of major axis 2a and minor
axis 2b , where a  b , which touches the X axis, is _____.
Options:
(a) ( xy 2 – b 2 x ) + ( y 2 – b 2 ) = 4 ( a 2 – b 2 ) y 2
2 2
(b) ( xy 2 + b 2 x ) + ( y 2 – b 2 ) y 2 = 4 ( a 2 – b 2 ) y 4
2 2
(c) ( xy 2 + b 2 x ) – ( y 2 – b 2 ) = 4 ( a 2 – b 2 ) y 2
2 2
(d) ( xy 2 + b 2 x ) + ( y 2 + b 2 ) = 4 ( a 2 – b 2 ) y 2
2 2

Question 14: The locus of a point P moving in the XY plane, such that the ratio of its
distance from S(1, 2) and line x + y = 3 is equal to k , is _____.
Options:
(a) a hyperbola if k  1
(b) a pair of straight line for k  0
(c) an ellipse if k  1
(d) a parabola if k = 1

https://drive.google.com/file/d/1hb41Ai40LxsEgyCyg3Pna6147nC-G56R/view?usp=sharing
Question 15: The equation of the transverse axis of a hyperbola which passes through ( 2, 4 )
and whose asymptotes are given by the equation ( x − y )( x + y + 1) = 0 is _____.
Options:
(a) x + y = 0
1
(b) y = −
2
1
(c) x = −
2
(d) none of these

https://drive.google.com/file/d/1cwK2ETDDk4FR-RnBp2wvnvCNrlH8nkPs/view?usp=sharing
Question 16: The maximum number of common normals of y = 4ax and

2
x 2 = 4by is equal
to _____.
Options:
(a) 3
(b) 4
(c) 5
(d) 6

Question 17: If the variable line y = kx + 2h is a tangent to the ellipse 2 x + 3 y = 6 , then the
2 2
locus of P ( h, k ) is a conic C, whose eccentricity equals _____.

Options:
5
(a)
2
7
(b)
3
7
(c)
3
(d) 2

https://drive.google.com/file/d/1aaIDP-g9YzNZMXuBCszms9D0-xVZyntN/view?usp=sharing
x2 y 2
Question 18: Consider an ellipse E  + − 1 = 0 . Suppose that C is any circle concentric
16 4
with E. Let A be a point on E and let B be a point on C, such that AB is tangential to both E
and C.
The maximum length of AB is _____.
Options:
(a) 2
(b) 4
(c) 2 2
(d) 4 2
https://drive.google.com/file/d/1h7l4-zrOXQE5rgp-m0p4JTbuL8hiO2dY/view?usp=sharing
x2 y 2
16 4
and C.
The slope of the tangent AB, when the length of AB is maximum is _____.
Options:
(a) 2
1
(b)
2
3
(c)
2
2
(d)
3

x2 y 2
16 4
and C.
The radius of circle C, when length of AB is maximum, is _____.
Options:
(a) 4 2
(b) 3 2
(c) 2 2
(d) 4

Question 21: A conic passes through the point ( 2, 4 ) and is such that the segment of any of
its tangents at any point contained between the coordinate axes is bisected at the point of
tangency.
Its eccentricity is _____.
Options:
(a) 2
(b) 2
(c) 3
3
(d)
2

tangency.
The foci of the conic are _____.
Options:
( ) (
(a) 2 2, 0 and −2 2, 0 )
(b) ( 2 ) (
2, 2 2 and −2 2, − 2 2 )
(c) ( 4, 4 ) and ( −4, − 4)
( ) (
(d) 4 2, 4 2 and −4 2, − 4 2 )
tangency.
The equation of its directrices are _____.
Options:
(a) x + y = 8
(b) x + y = 4
(c) x + y = 4 2
(d) none of these

x2 y 2
Question 24: Let S, S be the foci of the ellipse + = 1 whose eccentricity is e. P is a
a 2 b2
variable point on the ellipse. Consider the locus of the incentre of the  PSS .
The locus of the incentre is _____.
Options:
(a) an ellipse
(b) a hyperbola
(c) a parabola
(d) a circle

https://drive.google.com/file/d/1IbenxXBBe1WfwgDyoHvumgF3xIPc9IPV/view?usp=sharing
x2 y 2
Question 25: Let S, S be the foci of the ellipse + = 1 whose eccentricity is e. P is a
a 2 b2
The eccentricity of the locus of P is _____.
Options:
2e
(a)
1− e
2e
(b)
1+ e
(c) 1
(d) none of these

x2 y 2
Question 26: Let S, S be the foci of the ellipse 2 + 2 = 1 whose eccentricity is e. P is a
a b
The maximum area of the rectangle inscribed in the locus is _____.
Options:
2abe2
(a)
1+ e
2abe
(b)
1− e
abe
(c)
1+ e
(d) none of these

Question 27: If Sk has minimum possible radius equal to that of S1 and Sm , distance
between whose centres is a  PQ , then a2 is equal to _____.
Options:
(a) 3
(b) 4
(c) 2
(d) 9

https://drive.google.com/file/d/1dopZPT77OuCxHR2eicyoXHDqDA5vSeht/view?usp=sharing
Question 28: An ellipse whose major axis is parallel to X –axis such that the segments of the
focal chords are 1 and 3 units. The lines a x + b y + c = 0 are the chords of the ellipse such that
a, b, c are in A.P. and bisected by the point at which they intersect. The equation of its
auxiliary circle is x 2 + y 2 + 2  x + 2  y − 2 − 1 = 0 then _____.
Equation of the auxiliary circle is
Options:
(a) x 2 + y 2 − 2 x + 4 y + 1 = 0
(b) x 2 + y 2 + 2 x + 2 y − 3 = 0
(c) x2 + y 2 + 2 x + 4 y + 1 = 0
(d) x2 + y 2 − 4 x + 2 y − 3 = 0

https://drive.google.com/file/d/1x4YgFs18Dj_rQci37_RK6Ws1gZ-YUAER/view?usp=sharing
Question 29: Consider a parabola (P) having focus at F (1, 2 ) and touching both the
coordinate axes. Equation of directrix of parabola p is _____.
Options:
(a) y + x = 0
(b) y + 2x = 0
(c) y + 3x = 0
(d) y + 4x = 0

https://drive.google.com/file/d/1ALSu3XRxE92iqHJrGQphrR5IWTmcfneQ/view?usp=sharing
Question 30: Consider a parabola (P) having focus at F (1, 2 ) and touching both the
coordinate axes. From point R on axis of parabola P, three real and distinct normals are
drawn to the parabola, then coordinate of R will be _____.
Options:
12
(a) ( 2 − 3,  ) /  
5
12
(b) ( 2 − 3,  ) /  =
5
12
(c) ( 2 − 3,  ) / 0   
5
(d) ( 2 − 3,  ) /  

Question 31: A straight line L is drawn through the origin and parallel to the tangent to the
curve f ( x, y ) = 0 at an arbitrary point M on the curve. P is the point of intersection of the
line L and the straight line parallel to the Y–axis and passing through the point M. If
f ( x, y )  y − log b x then the locus of P is a _____.
Options:
(a) Straight line
(b) Parabola
(c) Circle
(d) Central conic

https://drive.google.com/file/d/1Ew8YhzIbR9ZWJEtmRi9yMQTvSP29cHwT/view?usp=sharing
line L and the straight line parallel to the Y–axis and passing through the point M. .If
f ( x, y )  y2 − 4 a x then the locus of P is a _____.
Options:
(a) Straight line
(b) Parabola
(c) Circle
(d) Central conic

line L and the straight line parallel to the Y–axis and passing through the point M. If
 a + a2 − x2 
f ( x, y )  y − a 2 − x 2 + a ln   then the locus of P is a _____.
 x 
 
Options:
(a) Straight line
(b) Parabola
(c) Circle
(d) Central conic

Question 34: From a point P, common tangents are drawn to the ellipse x 2 + 4 y 2 = 8 and
parabola y 2 = 4 x . The equation of the common tangents to the ellipse and the parabola are
_____.
Options:
(a) x − 3 =  2 y
(b) x + 4 =  2 y
(c) x − 4 =  2 y
(d) x + 3 =  2 y

MCQ
Question 35: A circle externally touches the two circles x 2 + y 2 = 1 and x 2 + y 2 = 4 x . The
locus of the centre of the circle is a conic C and e is its eccentricity. Then, _____.
Options:
(a) C is an ellipse
(b) C is a hyperbola
(c) e = 2
(d) e2 = 2

https://drive.google.com/file/d/1ROlsRT0s7MLf1vZioiSxnOYnR7sajI3h/view?usp=sharing
Question 36: If a parabola touches the lines y = x and y = − x at P ( 3, 3) and Q ( 2, − 2)

respectively, then _____.
Options:
 30 −6 
(a) focus is  , −  .
 13 13 
(b) equation of directrix is 5x + y = 0 .
(c) equation of line through origin and focus is x + 5 y = 0 .
(d) equation of line through origin and parallel to axis is x − 5 y = 0 .

https://drive.google.com/file/d/1_qVVR5I7aZn4dy74dFCWicv2au1--2qV/view?usp=sharing
Question 37: A hyperbola has a focus at origin, its eccentricity is 2 and the corresponding
directrix is x + y + 1 = 0 . The equation of its asymptotes is/are _____.
Options:
(a) x + 1 = 0
(b) x − 1 = 0
(c) y + 1 = 0
(d) y − 1 = 0

https://drive.google.com/file/d/1nLkakTGIXlRpfu_1HUaNyGnay09qNFqg/view?usp=sharing
Question 38: If the tangents at the points A( x1 , y1 ) and B( x2 , y2 ) to the parabola y 2 = 4 x

intersect at C ( x3 , y3 ) , and l1 , l2 , l3 are the lengths of the perpendiculars on any of the
tangents of the given parabola from the points A, B, C respectively, then _____.
Options:
(a) x1 , x3 , x2 are in GP
(b) y1 , y3 , y2 are in AP
(c) l1 , l3 , l2 are in HP
(d) l1 , l3 , l2 are in GP

https://drive.google.com/file/d/1C46gwtuZNLEOGqL6HEN97O3Gj7zaCXpp/view?usp=sharing
Question 39: Let A, B and C be three distinct points on y 2 = 8 x such that the normals at
these points are concurrent at P. The slope of AB is 2 and the abscissa of the centroid of 
4
ABC is . Which of the following is (are) correct?
3
Options:
(a) area of ABC is 8 square units
(b) coordinates of P are (6,0)
(c) angle between normals are 45°, 45°, 90°
(d) angle between normals are 30°, 30°, 60°

Question 40: A hyperbola centred at C has one focus at P ( 6, 8) . If its directrices are
3x + 4 y + 10 = 0 and 3x + 4 y –10 = 0 , then _____.
Options:
(a) CP = 10
(b) eccentricity = 5
(c) CP = 8
5
(d) eccentricity =
2

https://drive.google.com/file/d/1ZOU_37HIdDKINpIzj-M8NnDTnIGp_nr-/view?usp=sharing
Question 41: Consider the statement regarding a conic : “If PSQ be a focal chord, and X the
foot of the corresponding directrix, such that XP and XQ are equally inclined to the axis of a
conic”. The conics which always satisfy this property is/are _____.
Options:
(a) Parabola
(b) Ellipse
(c) Hyperbola
(d) Rectangular Hyperbola

https://drive.google.com/file/d/1tRqinT8xe79kE4-m1aVUl8b7tfIYl8RF/view?usp=sharing
Question 42: Let L be the point ( t , 2) and M be a point on the y axis such that LM has slope
− t . Then the locus of the midpoint of LM. as t varies over all real values, is a parabola,
whose.
Options:
(a) Vertex is ( 0, 2 )
(b) Latus – rectum is of length 2
 17 
(c) Focus is  0, 
 8 
(d) Directrix is 8 y −15 = 0

https://drive.google.com/file/d/1q-OfmhfmCkyqvzhy7Tmh2a-oBmODSkrH/view?usp=sharing
Question 43: Let A = ( −1, 0) and B = ( 2, 0) be two points on the x – axis. A point M is
moving in the xy – plane in such a way that  MBA = 2  MAB . Then the point M moves
along a conic whose
Options:
(a) Eccentricity is 2
1
(b) Eccentricity is
2
(c) Latus – rectum is of length 6
1
(d) Directrices are x =   
2

https://drive.google.com/file/d/1ROlsRT0s7MLf1vZioiSxnOYnR7sajI3h/view?usp=sharing
x2 y2 x2 y2  
Question 44: Two ellipses + = 1 and + =10   
cos  sin 
2 2
sin  cos 
2 2
 4
intersect at four points P,Q, R, S then which of the following statement(s) is /are true ?
Options:
(a) PQRS is a square with length of the side sin 2
sin 2
(b) PQRS lie on a circle whose centre is origin and with radius
2
(c) eccentricity of the two given ellipses are same
x2 y2
(d) there are two points on + = 1 whose reflection in y = x lie on the same
sin 2  cos 2 
ellipse

https://drive.google.com/file/d/1Guc1Ys9oJaQzpPwxvi_7CMY350KWyjkU/view?usp=sharing
INTEGER
x2 y 2
Question 45: The number of points on the hyperbola − = 1 from where mutually
a 2 b2
perpendicular tangents can be drawn to circle x 2 + y 2 = a 2 is _____.

https://drive.google.com/file/d/16w6KBs6ba73pOZo5J9-CFba1KMtqnlE-/view?usp=sharing
Question 46: If p is the length of the perpendicular from a focus upon the tangent at any
x2 y 2 2a b 2
point P on the ellipse 2 + 2 = 1 and r is the distance of P from the focus, then − 2 =
a b r p
_____.

x2 y 2
Question 47: If a tangent of slope 2 of the ellipse + = 1 is normal to the circle
a 2 b2
x 2 + y 2 + 4 x + 1 = 0 , then maximum value of ab is _____.

Question 48: If the length of the latus rectum of a standard hyperbola of eccentricity 2 is
3 5 7
equal to the limit of the series 2 + 2 + 2 + ... and r is the radius of the director
1 1 + 2 1 + 22 + 32
2
circle of its conjugate hyperbola, then r 2 = _____.

https://drive.google.com/file/d/1IjX6YI_cSts16o8oIFdSaXTXFfvJy8cl/view?usp=sharing
96
Question 49: Let P, Q be points on the ellipse 16 x 2 + 25 y 2 = 400 , so that PQ =and P, Q
25
lie above the major axis. The circle drawn with PQ as diameter touches the major axis at the
1
positive focus. If m is the slope of PQ, then find the value of .
m

https://drive.google.com/file/d/1Guc1Ys9oJaQzpPwxvi_7CMY350KWyjkU/view?usp=sharing
Question 50: The normal chord at a point t on the parabola y 2 = 4ax subtends a right angle
at its vertex. Find the value of t 2 .

https://drive.google.com/file/d/1E7E471u_nxRMl9zEIIrvgKP2YKKrOTa-/view?usp=sharing
Question 51: A parabola with directrix x + y + 2 = 0 touches a line 2 x + y – 5 = 0 at ( 2, 1) .

m
If m is the length of the latus rectum of the parabola, then the value of is _____.
2

https://drive.google.com/file/d/1UljIWZm1l5Rr7Squ3MU1WckWD1o9MK_7/view?usp=sharing
Question 52: Let C1 : x 2 − y 2 = 5 and C2 : x 2 + y 2 − 8 y + 3 = 0 be the equations of a

hyperbola and a circle respectively. The curves C1 and C 2 touch each other at ( 3,2) .
( )
P 0, 4 − 13 is a point on the curve C 2 . Let a line through P meet C1 at m number of
points and C 2 at n number of points. If ( m + n ) = 3 , then the number of such straight lines is
_____.

https://drive.google.com/file/d/1BOSLWOzKQr18Cps7qhpzVO6o0U5bhy3f/view?usp=sharing
Question 53: The number of circles of the form x 2 + y 2 = r 2 that can be drawn, such that
they neither touch nor intersect the curve xy = 8 , where r  is _____.

Question 54: From the origin, tangents OA and OB are drawn to the curve
( x – 2) + ( y – 2) = 1. If the line PQ, where P and Q are respectively the midpoints of OA
2 2
and OB, touches the curve ( y + 3) = 4a ( x + 4 ) and the length of latus rectum of the
2
l
parabola is l , then is _____.
7

https://drive.google.com/file/d/1C46gwtuZNLEOGqL6HEN97O3Gj7zaCXpp/view?usp=sharing
Question 55: If two distinct chords of a parabola y 2 = 4ax passing through ( a,2a ) are
bisected by the line x + y = 1, and 4a is a natural number, then the maximum length of the
latus–rectum is _____.

https://drive.google.com/file/d/18C00JCqWZlNaBnL9xOnSGozBxxUInmp7/view?usp=sharing
Question 56: Area of trapezium whose vertices lie on the parabola y 2 = 4 x and its diagonals
pass through (1, 0) and having length
25 75
unit each, is . Then  is _____.
4 

Question 57: Coordinates of the vertices B and C are ( 2, 0 ) and (8, 0) respectively. The
B C
vertex A is varying in such a way that 4 tan tan = 1 . If the locus of A is an ellipse then
2 2
the length of its semi major axis is _____.

Question 58: A parabola is drawn through two given points A (1, 0) and B( −1, 0) such that
its directrix always touches the circle x 2 + y 2 = 4. If the maximum possible length of semi
latus–rectum is k then  k  is (where [.] denotes greatest integer function)

https://drive.google.com/file/d/1q-OfmhfmCkyqvzhy7Tmh2a-oBmODSkrH/view?usp=sharing
Question 59: Normals drawn through the point P ( 2, 3) intersects the curve xy = 4 at
A ( x1, y1 ) , B( x2 , y2 ) , C ( x3 , y3 ) , D ( x4 , y4 ) . The value of
1 1 1 1 
( x1 + x2 + x3 + x4 )  + + +  is _____.
 y1 y2 y3 y4 

https://drive.google.com/file/d/1NllVl_qUqT--zdaBLOrIyBJgJmqTTXaA/view?usp=sharing
MATRIX
Question 60: Match the following.
Column – I Column – II
(A) If three normals can be drawn to the curve y 2 = x from the point (p) 0
( c, 0) , then c can be _____.
(B) If the sides and the angles of a plane triangle vary in such a way (q) 1
da db dc
that its circumradius remains constant, then + + =
cos A cos B cos C
_____.
(C) If the point ( a, a ) lies between the lines x + y = 6 , then  a  is (r) 2
_____, where [.] denotes the greatest integer function and |.| denotes
the absolute value function)
(D) The point ( p + 1 ,  p ) lies inside the region bounded by the (s) 3
circles x 2 + y 2 − 2 x − 15 = 0 and x 2 + y 2 − 2 x − 7 = 0 . The number of
possible values of p is _____, where [.] denotes the greatest integer
function and |.| denotes the absolute value function).
(t) 4
Options:
(a) A → q, r, s, t; B → p; C → p, q, r; D → p
(b) A → q, r, s, t; B → p; C → p, r; D → p
(c) A → q, t; B → p; C → p, q, r; D → p
(d) A → q, r, t; B → p; C → p, q, r; D → p

Question 61: Match the following:

(a) The normal chord at a point on (p) 4
the
parabola y 2 = 4 x subtends a right
angle at the
vertex,
(b) The area of the triangle inscribed in the (q) 2
curve y 2 = 4 x , the parameter of coordinates
whose vertices are 1, 2 and 4 is
(c) The number of distance normal possible (r) 3
 11 1 
form  ,  to the parabola y 2 = 4 x is
 4 4
(d) The normal at ( a, 2a ) on y 2 = 4ax (s) 6
meets the curve again at ( at 2 , 2at ) , then the
value of t − 1 is
Options:
(a) (a) →(p); (b) → (s); (c) → (q); (d) → (p)
(b) (a) →(q); (b) → (r); (c) → (q); (d) → (p)
(c) (a) →(q); (b) → (s); (c) → (q); (d) → (r)
(d) (a) →(q); (b) → (s); (c) → (q); (d) → (p)


(a) The parabola y = ax 2 + bx + c has vertex (p) 1
( p, p ) any y - intercept ‘ − p ’, where
p  0 . The value of ‘ b ’ is not less than
(b) Two of the altitudes of a scalene triangle (q) 2
ABC have length 4 and 12. If the length of
third altitude is ‘ p ’ it is also an integer,
then ‘ p ’ can be
(c) The number of points in the complex (r) 3
plane satisfying z − 4 − 8i = 10 and
z − 3 − 5i + z − 5 − 11i = 4 5 is not more
than
(d) The straight line 3x + 4 y − 12 = 0 (s) 4
x2 y 2
intersects the ellipse + = 1 at two
16 9
points A and B. There is a point p on the
ellipse such that area of  PAB is 3. The
number of such points ‘ p ’ is not greater
than
(t) 5
Options:
(a) a → p, q, r, s; b → s, t; c → q, r, s, t; d → q, r, s, t
(b) a → p, q, s; b → s, t; c → q, r, s, t; d → q, r, s, t
(c) a → p, q, r, s; b → s, t; c → q, t; d → q, r, s, t
(d) a → p, q, r, s; b → s; c → q, r, s, t; d → q, r, s, t

https://drive.google.com/file/d/150d8bLYcFtBVheziYG1D1Mf22HJKIUXy/view?usp=sharing

(a) No of points of intersection of curves (p) 1
y = ln x and ( x − 1) + y 2 − 4 = 0 .
2
S  (q) 2
n
(b) Sn =  r !( n  6 ) then Sn − 7  n  is
r =1 7
(where [ x ] denotes integer less than or
equal to x )
(c) Equation of tangent z0 to the circle (r) 3
 z
z = r is Re   =  , then  is
 z0 
(d) Normal drawn at any point of an ellipse (s) 5
x2 y 2
+ = 1 , is tangent to the circle
25 9
x 2 + y 2 = r 2 then maximum value of r is
(t) 5
Options:
(a) a → s; b → s, t; c → p; d → q
(b) a → r; b → s, t; c → p; d → q
(c) a → r; b → s; c → p; d → q
(d) a → r; b → s, t; c → s; d → q

https://drive.google.com/file/d/18gjIcUYiw7wmn54EI5_egRfFAexz_8u-/view?usp=sharing

(a) The straight line joining the points (p) – 4
( 0, 3) and (5, − 2) is a tangent to the curve
C
y= then C =
x +1
(b) For each parabola y = x 2 + px + q ( p , q (q) 2
are parameters meeting) coordinate axes at 3
distinct points, if circles are drawn through
these points, and the family of circle passes
through the fixed point ( h, k ) then
h + 4k =
(c) If OB is the one of the rhombus of area (r) 0
2 3 sq. units where O = ( 0, 0) and
B = ( 2, 0) and the other vertices are
( x1, y1 ) , ( x2 , y2 ) in the first quadrant then
x1 + x2 + y1 − y2 =
(d) The distance between the lines (s) 4
( x + 7 y ) + 4 2 ( x + 7 y ) − 42 = 0 .
2
Options:
(a) (a) → (q); (b) → (s); (c) → (s); (d) → (q)
(b) (a) → (s); (b) → (p); (c) → (s); (d) → (q)
(c) (a) → (s); (b) → (s); (c) → (s); (d) → (q)
(d) (a) → (r); (b) → (s); (c) → (s); (d) → (q)

https://drive.google.com/file/d/1dopZPT77OuCxHR2eicyoXHDqDA5vSeht/view?usp=sharing
https://drive.google.com/file/d/1ormF4c1tkueSTHIPsLp7i7BoZd5eQejN/view?usp=sharing

4
For the ellipse ( 3 x – 6 ) + ( 3 y – 9 ) = ( 5 x + 12 y + 6 )
2 2 2
169
(a) Length major axis 24
(p)
5
(b) Length minor axis 16
(q)
5
(c) Length of Latus Rectum 16
(r)
3
(d) Distance between directrices 72
(s)
5
48
(t)
5
Options:
(a) a → r; b → q; c → r; d → s
(b) a → t; b → q; c → r; d → s
(c) a → t; b → s; c → r; d → s
(d) a → t; b → q; c → q; d → s

PARAGRAPH
Question 66: From a point P on the parabola y 2 = 4ax , normals at the points Q and R, are
drawn to the same parabola. Now, a circle is drawn to circumscribe the  PQR. If the
circumcircle of  PQR cuts the parabola at another point M, then,
Statement I : The common chords QR and PM intersect each other at the directrix.
Statement II : The chord of contact of the parabola with respect to any point lying on
the directrix passes through the focus of the parabola.
Options:
(a) Statement I is True; Statement II is True; Statement II is a correct explanation
for Statement I.
(a) Statement I is True; Statement II is True; Statement II is not a correct explanation
for Statement I.
(c) Statement I is True; Statement II is False.
(d) Statement I is False; Statement II is True.

Question 67:
Statement I : The locus of a point P, with respect to which, the chord of contact of the curve
y 2 = 4ax subtends 90° at origin, is x = −4a .
because
Statement II : If the normal drawn at the point t1 on the curve y 2 = 4ax cuts the parabola
again at t 2 if t1 is real, when t2  2 2 .
Options:
(a) Statement I is True; Statement II is True; Statement II is a correct explanation
for Statement I.
(a) Statement I is True; Statement II is True; Statement II is not a correct explanation
for Statement I.
(c) Statement I is True; Statement II is False.
(d) Statement I is False; Statement II is True.

Question 68: Statement –I : If the circles x 2 + y 2 − 2 x + 4 y + 4 = 0 and

x 2 + y 2 − 4 x − 2 y + c = 0 intersect such that the common chord is longest, when c = −4 .
Statement – II : If two circles intersect, then the common chord so obtained, is longest when
it is a diameter of the smaller circle.
Options:
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct Explanation for
Statement-1
(b) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation
for Statement-1
(c) Statement-1 is True, Statement-2 is False
(d) Statement-1 is False, Statement-2 is True

https://drive.google.com/file/d/1bBlhGl07H395Y4C-X35VLdVtR-JYe7t2/view?usp=sharing
Question 69: Statement 1: The equation x 2 + 2  x y + y 2 + 2 x + 2 y + 4 = 0 represents an

ellipse
if  ( −1, 1) .
Statement 2: The general equation of second degree represents an ellipse if   0 ,
h2 − ab  0
Options:
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct Explanation for
Statement-1
(b) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation
for Statement-1
(c) Statement-1 is True, Statement-2 is False
(d) Statement-1 is False, Statement-2 is True

https://drive.google.com/file/d/1tRqinT8xe79kE4-m1aVUl8b7tfIYl8RF/view?usp=sharing

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