Conic Sections

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

If the circles (x+1)2+(y+2)2=r2 and x2+y2-4x-4y+4=0 intersect at exactly two distinct points, then

1. 12<r<7
2. 3<r<7
3. 5<r<9
4. 0<r<7