Conic Sections

If the points of intersection of two distinct conics x2+y2=4b and x216+ y2b2=1 lie on the curve y2=3x2, then 3 3 times the area of the rectangle formed by the intersection points is .

Let P( , ) be a point on the parabola y2=4x. If P also lies on the chord of the parabola x2=8y whose mid point is (1, 54 ), then ( -28)( -8) is equal to .

If the shortest distance of the parabola y2=4x from the centre of the circle x2+y2-4x-16y+64=0 is d, then d2 is equal to :

1. 16
2. 24
3. 20
4. 36

The length of the chord of the ellipse x225+ y216=1, whose mid point is (1, 25 ), is equal to :

1. 16915
2. 20095
3. 15415
4. 17415

Four distinct points (2k,3k),(1,0),(0,1) and (0,0) lie on a circle for k equal to :

1. 313
2. 213
3. 513
4. 113

Let e1 be the eccentricity of the hyperbola x216- y29=1 and e2 be the eccentricity of the ellipse x2a2+ y2b2=1, a>b, which passes through the foci of the hyperbola. If e1e2=1, then the length of the chord of the ellipse parallel to the x-axis and passing through (0,2) is :

1. 8 53
2. 3 5
3. 4 5
4. 10 53

Consider a circle (x- )2+(y- )2=50, where , >0. If the circle touches the line y+x=0 at the point P, whose distance from the origin is 4 2, then ( + )2 is equal to .