Conic Sections

For 0< < /2, if the eccentricity of the hyperbola x2-y2 cosec2 =5 is 7 times eccentricity of the ellipse x2 cosec2 +y2=5, then the value of is :

1. 6
2. 5 12
3. 3
4. 4

Let x2a2+ y2b2=1, a>b be an ellipse, whose eccentricity is 1 2 and the length of the latus rectum is 14. Then the square of the eccentricity of x2a2- y2b2=1 is :

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

Let C: x2+y2=4 and C': x2+y2-4 x+9=0 be two circles. If the set of all values of so that the circles C and C' intersect at two distinct points, is R-[a, b], then the point (8a+12, 16b-20) lies on the curve :

1. x2+2y2-5x+6y=3
2. 5x2-y=-11
3. x2-4y2=7
4. 6x2+y2=42

Let the line L: 2x+y= pass through the point of the intersection P (in the first quadrant) of the circle x2+y2=3 and the parabola x2=2y. Let the line L touch two circles C1 and C2 of equal radius 2 3. If the centres Q1 and Q2 of the circles C1 and C2 lie on the y-axis, then the square of the area of the triangle PQ1Q2 is equal to .

Let P be a point on the ellipse x29+ y24=1. Let the line passing through P and parallel to y-axis meet the circle x2+y2=9 at point Q such that P and Q are on the same side of the x-axis. Then, the eccentricity of the locus of the point R on PQ such that PR: RQ=4: 3 as P moves on the ellipse, is :

1. 1321
2. 13923
3. 137
4. 1119

Let the locus of the midpoints of the chords of the circle x2+(y-1)2=1 drawn from the origin intersect the line x+y=1 at P and Q. Then, the length of PQ is :

1. 12
2. 1
3. 1 2
4. 2

If one of the diameters of the circle x2+y2-10x+4y+13=0 is a chord of another circle C, whose center is the point of intersection of the lines 2x+3y=12 and 3x-2y=5, then the radius of the circle C is :

1. 4
2. 3 2
3. 6
4. 20

If the foci of a hyperbola are same as that of the ellipse x29+ y225=1 and the eccentricity of the hyperbola is 158 times the eccentricity of the ellipse, then the smaller focal distance of the point ( 2, 143 25 ) on the hyperbola, is equal to

1. 14 25- 43
2. 7 25+ 83
3. 7 25- 83
4. 14 25- 163

Let the foci and length of the latus rectum of an ellipse x2a2+ y2b2=1, a>b be ( 5, 0) and 50, respectively. Then, the square of the eccentricity of the hyperbola x2b2- y2a2b2=1 equals

Let P be a parabola with vertex (2,3) and directrix 2x+y=6. Let an ellipse E: x2a2+ y2b2=1, a>b, of eccentricity 1 2 pass through the focus of the parabola P. Then, the square of the length of the latus rectum of E, is

1. 51225
2. 65625
3. 3858
4. 3478