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If the distance between the foci and the distance between the directrices of the hyperbola (x² / a²) – (y² / b²) = 1 are in the ratio 3 : 2, then a : b is

1) √2 : 1 2) √3 : √2 3) 1 : 2 4) 2 : 1 Solution: (1) √2 : 1 Given 2ae / [2a / e] = 3 / 2 ⇒ e2 = 3 / 2 ⇒ (a2 + b2) / a2 = 3 / 2 ⇒ (b / a)2 =... View Article

Let P (a sec θ, b tan θ) and Q (a sec ɸ, b tan ɸ), where θ + ɸ = π / 2, be 2 points on the hyperbola (x² / a²) – (y² / b²) = 1. If (h, k) is the point of intersection of the normals at P and Q, then k =

1) [a2 + b2] / a 2) - [a2 + b2] / a 3) [a2 + b2] / b 4) - [a2 + b2] / b Solution: (4) - [a2 + b2] / b The slope of normal to (x2 / a2) - (y2... View Article

A focus of an ellipse is at the origin. The directrix is the line x = 4 and the eccentricity is 1/2, the length of the semi-major axis is

1) 5/3 2) 8/3 3) 2/3 4) 4/3 Solution: (2) 8/3 Directrix = |(a / e) - ae| = 4 (a / e) - ae = 4 a [(1 / e) - e] = 4 a [2 - (1 / 2)] = 4 a * (3... View Article

A parabola has the origin as its focus and the line x = 2 as the directrix. Then, the vertex of the parabola is at

1) (2, 0) 2) (0, 2) 3) (1, 0) 4) (0, 1) Solution: (3) (1, 0) The vertex is a midpoint of focus and directrix (1, 0) is the vertex.... View Article

If I denotes the semi – latus rectum of the parabola y² = 4ax and SP and SQ denote the segments of any focal chord PQ, S being the focus, then SP, I and SQ are in the relation

1) AP 2) GP 3) HP 4) I2 = SP2 + SQ2 Solution: (3) HP Let 2 points on the parabola be P (at12, 2at1) and Q = (at22, 2at2) SP = √(a - at12)2... View Article

If a hyperbola has a length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricity of the hyperbola is :

1) 9 2) 13 / 12 3) 32/3 4) 64/3 Solution: (2) 13 / 12 2b = 5 2ae = 13 ae = 13 / 2 a2e2 = a2 + b2 a = b e2 = 1 + (b2 / a2) e = 13 / 12... View Article

The equation y² – 8y – x + 19 = 0 represents

1) a parabola whose focus is (1/4, 0) and directrix is x = -1/4 2) a parabola whose vertex is (3, 4) and directrix is x = 11/4 3) a parabola... View Article

Equation of the directrix of parabola 2x² = 14y is equal to

1) y = - (7/4) 2) x = - (7/4) 3) y = (7/4) 4) x= (7/4) Solution: (1) y = - (7/4) The equation of the directrix parabola 2x2 = 14y x2 = 7y a... View Article

In an ellipse, if the lines joining focus to the extremities of the minor axis from an equilateral triangle with the minor axis, then the eccentricity of the ellipse is

1) √3/2 2) √3/4 3) 1/√2 4) √2 / 3 Solution: (1) √3 / 2 Consider the ellipse (x2 / a2) + (y2 / b2) = 1 where a2 > b2 BB’S forms an... View Article

If the distance between the directrices of a rectangular hyperbola are 10, then the distance between its foci will be

1) 10√2 2) 5 3) 5√2 4) 20 Solution: (4) 20 Distance between directrices = 2a / e. ∵ The eccentricity of rectangular hyperbola = √2. Distance... View Article

The coordinates of the focus of the parabola described parametrically by x = 5t² + 2, y = 10t + 4 are

1) (7, 4) 2) (3, 4) 3) (3, -4) 4) (-7, 4) Solution: (1) (7, 4) x = 5t2 + 2, y = 10t + 4 (x - 2) / 5 = t2 and [y - 4] / 10 = t (x - 2) / 5 =... View Article

If e₁ is the eccentricity of the ellipse (x² / 16) + (y² / 7) = 1 and e₂ is the eccentricity of the hyperbola (x² / 9) – (y² / 7) = 1, then e₁ + e₂ =

1) 16/7 2) 25/4 3) 25/12 4) 16/9 5) 23/16 Solution: (3) 25/12 The equation of the ellipse (x2 / a2) + (y2 / b2) = 1. (x2 / 16) + (y2 / 7) =... View Article

The sum of the distance of a point (2, -3) from the foci of an ellipse 16(x – 2)² + 25(y + 3)² = 400 is

1) 8 2) 6 3) 50 4) 32 5) 10 Solution: (2) 6 16(x - 2)2 + 25(y + 3)2 = 400 The equation can be rewritten as (x - 2)2 / 25 + (y + 3)2 / 16 =... View Article

The distance of a point P on the parabola y² = 12x, if the ordinate of P is 6, is

1) 12 2) 6 3) 3 4) 9 Solution: (2) 6 y2 = 12x a = 3 For point (x, y), y = 6 and x = 3.... View Article

S and T are the foci of an ellipse and B is the endpoint of the minor axis. If STB is an equilateral triangle, then the eccentricity of the ellipse is

1) 1/4 2) 1/3 3) 1/2 4) 2/3 Solution: (3) ½ STB is equilateral triangle. tan 60° = (OB / OS) = [b / {(1 / 2) 2ae}] --- as d(foci) = 2ae... View Article

The equation [x² / (1 – r)] – [y² / (1 + r)] = 1, r > 1 represents the

1) an ellipse 2) a hyperbola 3) a circle 4) none of these Solution: (4) none of these [x2 / (1 - r)] - [y2 / (1 + r)] = 1 ----- as r > 1... View Article

If the length of the major axis of an ellipse is 17/8 times the length of the minor axis, then the eccentricity of the ellipse is

1) 8/17 2) 15/17 3) 9/17 4) 2√2/17 5) 13/17 Solution: (2) 15/17 2a = (17 / 8) 2b a = (17 / 8) b b2 = a2 (1 - e2) (1 - e2) = (b2 / a2) = 64... View Article

If a point P (x, y) moves along the ellipse (x² / 16) + (y² / 25) = 1 and C is the centre of the ellipse, then the sum of maximum and minimum values of CP is

1) 25 2) 9 3) 4 4) 5 5) 16 Solution: (2) 9 (x2 / 16) + (y2 / 25) = 1 a2 = 16 and b2 = 25 a = 4 and b = 5 CP is maximum when P is at the end... View Article

One of the points on the parabola y² = 12x with focal distance 4 is

1) (3, 6) 2) (9, 6√3) 3) (7, 2√21) 4) (8, 4√6) Solution: (1) (3, 6) The coordinates of the focus of the parabola with equation y2 = 4ax are... View Article