Conic Sections

Let the length of a latus rectum of an ellipse x2a2 + y2b2 = 1 be 10. If its eccentricity is the minimum value of the function f(t) = t2 + t + 1112, t R, then a2 + b2 is equal to :

1. 115
2. 120
3. 125
4. 126

Let e1 and e2 be the eccentricities of the ellipse x2b2 + y225 = 1 and the hyperbola x216 - y2b2 = 1, respectively. If b < 5 and e1 e2 = 1, then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is :

1. 45
2. 35
3. 74
4. 32

Let p be the number of all triangles that can be formed by joining the vertices of a regular polygon P of n sides and q be the number of all quadrilaterals that can be formed by joining the vertices of P. If p + q = 126, then the eccentricity of the ellipse x216 + y2n = 1 is :

1. 1 2
2. 12
3. 74
4. 34

Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be 2a and 2b, respectively, and one focus and the corresponding directrix of this hyperbola be (-5, 0) and 5x + 9 = 0, respectively. If the product of the focal distances of a point ( , 2 5) on the hyperbola is p, then 4p is equal to .

The length of the latus-rectum of the ellipse, whose foci are (2, 5) and (2, -3) and eccentricity is 45, is

1. 503
2. 185
3. 65
4. 103

Let C be the circle x2 + (y - 1)2 = 2, E1 and E2 be two ellipses whose centres lie at the origin and major axes lie on x-axis and y-axis respectively. Let the straight line x + y = 3 touch the curves C, E1 and E2 at P(x1, y1), Q(x2, y2) and R(x3, y3) respectively. Given that P is the mid point of the line segment QR and PQ = 2 23, the value of 9(x1y1 + x2y2 + x3y3) is equal to .

Let the sum of the focal distances of the point P(4, 3) on the hyperbola H: x2a2 - y2b2 = 1 be 8 53. If for H, the length of the latus rectum is l and the product of the focal distances of the point P is m, then 9l2 + 6m is equal to :

1. 187
2. 184
3. 186
4. 185

Let for two distinct values of p the lines y = x + p touch the ellipse E: x242 + y232 = 1 at the points A and B. Let the line y = x intersect E at the points C and D. Then the area of the quadrilateral ABCD is equal to :

1. 48
2. 20
3. 24
4. 36

The centre of a circle C is at the centre of the ellipse E: x2a2 + y2b2 = 1, a > b. Let C pass through the foci F1 and F2 of E such that the circle C and the ellipse E intersect at four points. Let P be one of these four points. If the area of the triangle PF1F2 is 30 and the length of the major axis of E is 17, then the distance between the foci of E is :

1. 12
2. 26
3. 13
4. 132

The axis of a parabola is the line y=x and its vertex and focus are in the first quadrant at distances 2 and 2 2 units from the origin, respectively. If the point (1, k) lies on the parabola, then a possible value of k is :

1. 8
2. 3
3. 9
4. 4

The radius of the smallest circle which touches the parabolas y = x2 + 2 and x = y2 + 2 is

1. 7 216
2. 7 28
3. 7 22
4. 7 24

A line passing through the point P( 5, 5) intersects the ellipse x236 + y225 = 1 at A and B such that (PA) (PB) is maximum. Then 5 (PA2 + PB2 ) is equal to :

1. 290
2. 377
3. 338
4. 218

Let the product of the focal distances of the point P(4, 2 3) on the hyperbola H: x2a2 - y2b2 = 1 be 32. Let the length of the conjugate axis of H be p and the length of its latus rectum be q. Then p2 + q2 is equal to

Let C be the circle of minimum area enclosing the ellipse E : x2a2 + y2b2 = 1 with eccentricity 12 and foci ( 2, 0). Let PQR be a variable triangle, whose vertex P is on the circle C and the side QR of length 2a is parallel to the major axis of E and contains the point of intersection of E with the negative y-axis. Then the maximum area of the triangle PQR is :

1. 8(3 + 2)
2. 8(2 + 3)
3. 6(3 + 2)
4. 6(2 + 3)

The shortest distance between the curves y2 = 8x and x2 + y2 + 12y + 35 = 0 is:

1. 2 3 - 1
2. 2 2 - 1
3. 3 2 - 1
4. 2

If the four distinct points (4, 6), (-1, 5), (0, 0) and (k, 3k) lie on a circle of radius r, then 10k + r2 is equal to

1. 34
2. 32
3. 35
4. 33

If the equation of the hyperbola with foci (4, 2) and (8, 2) is 3x2 - y2 - x + y + = 0, then + + is equal to .

If S and S′ are the foci of the ellipse x218 + y29 = 1 and P be a point on the ellipse, then (SP S'P) + (SP S'P) is equal to :

1. 3(6 + 2)
2. 3(1 + 2)
3. 27
4. 9

Let the focal chord PQ of the parabola y2 = 4x make an angle of 60 with the positive x axis, where P lies in the first quadrant. If the circle, whose one diameter is PS, S being the focus of the parabola, touches the y-axis at the point (0, ), then 5 2 is equal to:

1. 15
2. 25
3. 20
4. 30

Let one focus of the hyperbola H : x2a2 - y2b2 = 1 be at ( 10, 0) and the corresponding directrix be x = 9 10. If e and l respectively are the eccentricity and the length of the latus rectum of H, then 9(e2 + l) is equal to :

1. 12
2. 14
3. 15
4. 16

The absolute difference between the squares of the radii of the two circles passing through the point (-9, 4) and touching the lines x + y = 3 and x - y = 3, is equal to .

Let the point P of the focal chord PQ of the parabola y2 = 16x be (1, -4). If the focus of the parabola divides the chord PQ in the ratio m : n, (m, n) = 1, then m2 + n2 is equal to :

1. 17
2. 37
3. 10
4. 26

If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is :

1. 3 19
2. 316
3. 4 17
4. 57

Let the ellipse E1 : x2a2 + y2b2 = 1, a > b and E2 : x2A2 + y2B2 = 1, A < B have same eccentricity 1 3. Let the product of their lengths of latus rectums be 32 3 and the distance between the foci of E1 be 4. If E1 and E2 meet at A, B, C and D, then the area of the quadrilateral ABCD equals :

1. 24 65
2. 18 65
3. 6 6
4. 12 65

Two parabolas have the same focus (4, 3) and their directrices are the x-axis and the y-axis, respectively. If these parabolas intersect at the points A and B, then (AB)2 is equal to :

1. 384
2. 392
3. 96
4. 192

Let the line x + y = 1 meet the circle x2 + y2 = 4 at the points A and B. If the line perpendicular to AB and passing through the mid-point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ABCD is equal to :

1. 14
2. 3 7
3. 2 14
4. 5 7

If x + y = 109 is the equation of the chord of the ellipse x29 + y24 = 1, whose mid point is ( 52, 12 ). then + is equal to :

1. 37
2. 46
3. 72
4. 58

Let a circle C pass through the points (4, 2) and (0, 2), and its centre lie on 3x + 2y + 2 = 0. Then the length of the chord, of the circle C, whose mid-point is (1, 2), is:

1. 4 2
2. 2 2
3. 2 3
4. 3

Let y2=12x be the parabola and S be its focus. Let PQ be a focal chord of the parabola such that (SP)(SQ)= 1474. Let C be the circle described taking PQ as a diameter. If the equation of a circle C is 64x2+64y2- x-64 3 y= , then - is equal to .

Let ABCD be a trapezium whose vertices lie on the parabola y2 = 4x. Let the sides AD and BC of the trapezium be parallel to y-axis. If the diagonal AC is of length 254 and it passes through the point (1, 0), then the area of ABCD is

1. 758
2. 1258
3. 252
4. 754