Conic Sections

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 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

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

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

Let the equation of the circle, which touches x-axis at the point (a, 0), a > 0 and cuts off an intercept of length b on y-axis be x2 + y2 - x + y + = 0. If the circle lies below x-axis, then the ordered pair (2a, b2) is equal to

1. ( , 2 + 4 )
2. ( , 2 - 4 )
3. ( , 2 - 4 )
4. ( , 2 + 4 )

If the midpoint of a chord of the ellipse x29 + y24 = 1 is ( 2, 4/3), and the length of the chord is 2 3, then is:

1. 26
2. 18
3. 22
4. 20

If A and B are the points of intersection of the circle x2 + y2 - 8x = 0 and the hyperbola x29 - y24 = 1 and a point P moves on the line 2x - 3y + 4 = 0, then the centroid of PAB lies on the line:

1. x + 9y = 36
2. 9x - 9y = 32
3. 4x - 9y = 12
4. 6x - 9y = 20