Conic Sections

Let the foci of a hyperbola coincide with the foci of the ellipse x236 + y216 = 1. If the eccentricity of the hyperbola is 5, then the length of its latus rectum is :

1. 96 5
2. 24 5
3. 12
4. 16

Let O be the vertex of the parabola x2 = 4y and Q be any point on it. Let the locus of the point P, which divides the line segment OQ internally in the ratio 2 : 3 be the conic C. Then the equation of the chord of C, which is bisected at the point (1, 2), is :

1. 5x - 4y + 3 = 0
2. x - 2y + 3 = 0
3. 4x - 5y + 6 = 0
4. 5x - y - 3 = 0

If the line x + 4y = 7, where R, touches the ellipse 3x2 + 4y2 = 1 at the point P in the first quadrant, then one of the focal distances of P is :

1. 1 3 - 12 11
2. 1 3 - 12 5
3. 1 3 + 12 5
4. 1 3 + 12 7

Let y2 = 12x be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the x-axis such that OPA = 90 . Then the locus of the centroid of such triangles OPA is:

1. y2 - 4x + 8 = 0
2. y2 - 2x + 8 = 0
3. y2 - 9x + 6 = 0
4. y2 - 6x + 4 = 0

Let one end of a focal chord of the parabola y2 = 16x be (16, 16). If P( , ) divides this focal chord internally in the ratio 5 : 2, then the minimum value of + is equal to:

1. 5
2. 7
3. 22
4. 16

Let the ellipse 3x2 + py2 = 4 pass through the centre C of the circle x2 + y2 - 2x - 4y - 11 = 0 of radius r. Let f1, f2 be the focal distances of the point C on the ellipse. Then 6f1f2 - r is equal to

1. 78
2. 68
3. 70
4. 74

Let r be the radius of the circle, which touches x-axis at point (a, 0), a < 0 and the parabola y2 = 9x at the point (4, 6). Then r is equal to .

Let C1 be the circle in the third quadrant of radius 3, that touches both coordinate axes. Let C2 be the circle with centre (1, 3) that touches C1 externally at the point ( , ). If ( - )2 = mn, (m, n) = 1, then m + n is equal to

1. 22
2. 13
3. 9
4. 31

Let P be the parabola, whose focus is (-2, 1) and directrix is 2x + y + 2 = 0. Then the sum of the ordinates of the points on P, whose abscissa is -2, is

1. 52
2. 32
3. 34
4. 14

Consider the hyperbola x2a2 - y2b2 = 1 having one of its foci at P(-3, 0). If the latus rectum through its other focus subtends a right angle at P and a2 b2 = 2 - , , N, then + is .