Conic Sections

Consider the parabola P : y2=4kx and the ellipse E : x2a2+ y2b2=1. Let the line segment joining the points of intersection of P and E, be their latus rectums. If the eccentricity of E is e, then e2+2 2 is equal to .

Let P(3 , 2 ), 0, be a point on the ellipse x29+ y24=1, Q be a point on the circle x2+y2-14x-14y+82=0 and R be a point on the line x+y=5 such that the centroid of the triangle PQR is (2+ , 3+ 23 ). Then the sum of the ordinates of all possible points R is :

1. 6
2. 2
3. 4
4. 8

Let H : x2a2- y2b2=1 be a hyperbola such that the distance between its foci is 6 and the distance between its directrices is 83. If the line x= intersects the hyperbola H at the points A and B such that the area of the triangle AOB is 4 15, where O is the origin, then 2 equals

1. 12
2. 16
3. 24
4. 25

Let A, B and C be the vertices of a variable right angled triangle inscribed in the parabola y2=16x. Let the vertex B containing the right angle be (4, 8) and the locus of the centroid of ABC be a conic C0. Then three times the length of latus rectum of C0 is .

Let an ellipse x2a2+ y2b2=1, a<b, pass through the point (4, 3) and have eccentricity 53. Then the length of its latus rectum is :

1. 4 53
2. 2 5
3. 7 53
4. 8 53

Let a circle C have its centre in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts from the coordinate axes. If the length of the chord of C on the line x+y=1 is 14, then the square of the radius of C is .

Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines x+(k-1)y+3=0 and 2x+k2y-4=0. If the line x-y+2=0 intersects the circle at the points A and B, then (AB)2 is equal to :

1. 10
2. 27
3. 18
4. 34

Let O be the origin, and P and Q be two points on the rectangular hyperbola xy=12 such that the midpoint of the line segment PQ is ( 12, - 12 ). Then the area of the triangle OPQ equals :

1. 32
2. 52
3. 72
4. 92

Let the parabola y=x2+px+q passing through the point (1, -1) be such that the distance between its vertex and the x-axis is minimum. Then the value of p2+q2 is :

1. 2
2. 4
3. 5
4. 8

Let A be the point (3, 0) and circles with variable diameter AB touch the circle x2+y2=36 internally. Let the curve C be the locus of the point B. If the eccentricity of C is e, then 72e2 is equal to .