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Axis of a parabola is y = x and vertex and focus are at a distance √2 and 2√2, respectively from the origin. Then, equation of the parabola is

1) (x - y)2 = 8 (x + y - 2) 2) (x + y)2 = 2 (x + y - 2) 3) (x - y)2 = 4 (x + y - 2) 4) (x + y)2 = 2(x - y + 2) Solution: (1) (x - y)2 = 8 (x... View Article

If e₁ is the eccentricity of the ellipse (x² / 16) + (y² / 25) = 1 and e₂ is the eccentricity of the hyperbola passing through the foci of the ellipse and e₁e₂ = 1, then the equation of the hyperbola is

1) (x2 / 9) - (y2 / 16) = 1 2)(x2 / 16) - (y2 / 9) = - 1 3) (x2 / 9) - (y2 / 25) = 1 4) None of these Solution: (2) (x2 / 16) - (y2 / 9) = -... View Article

If P be the point (1, 0) and Q a point on the locus of y² = 8x. The locus of mid – point of PQ is

1) x2 - 4y + 2 = 0 2) x2 + 4y + 2 = 0 3) y2 - 4x + 2 = 0 4) y2 + 4x - 2 = 0 Solution: (3) y2 - 4x + 2 = 0 Let point M (x’, y’) be the... View Article

The lines x = at² meets the ellipse (x² / a²) + (y² / b²) = 1, in the real points, iff

1) |t|< 2 2) |t| ≤ 1 3) |t|> 1 4) None of these Solution: (2) |t| ≤ 1 (x2 / a2) + (y2 / b2) = 1 Put x = at2 [a2t4 / a2] + [y2 / b2]... View Article

The curve represented by x = 3 (cos t + sin t) and y = 4(cos t – sin t) is

1) an ellipse 2) a parabola 3) a hyperbola 4) a circle Solution: (1) an ellipse x = 3(cos t + sint) y = 4(cos t - sint (x / 3)2 + (y / 4)2 =... View Article

The parametric representation of a point of the ellipse whose foci are (- 3, 0) and (9, 0) and eccentricity 1/3 is

1) (3 + 8 cos θ, 12 √2 sin θ) 2) (1 + 3 cos θ, 5 sin θ) 3) (1 + 3 cos θ, 1+ √5 sin θ) 4) (1 + 3 cos θ, 1 + 5 sin θ) 5) (1 + 3 cos θ, √5 sin... View Article

If (x² / 36) – (y² / k²) = 1 is a hyperbola, then which of the following statements can be correct?

1) (-3. 1) lies on the hyperbola 2) (3, 1) lies on the hyperbola 3) (10, 4) lies on the hyperbola 4) (5, 2) lies on the hyperbola Solution:... View Article

The equation of an ellipse whose eccentricity is 1/ 2 and the vertices are (4, 0) and (10, 0), is

1) 3x2 + 4y2 - 42x + 120 = 0 2) 3x2 + 4y2 + 42x + 120 = 0 3) 3x2 + 4y2 + 42x - 120 = 0 4) 3x2 + 4y2 - 42x - 120 = 0 Solution: (1) 3x2 + 4y2 -... View Article

The parametric representation of a point on the ellipse whose foci are ( -1, 0) and (7, 0) and eccentricity 1/2 is

1) (3 + 8 cos θ, 4 √3 sinθ) 2) ( 8 cos θ, 4 √3 sinθ) 3) (3 + 4 √3 sinθ, 8 sinθ) 4) None of the above Solution: (1) (3 + 8 cos θ, 4 √3 sinθ)... View Article

The curve described para metrically by x = t² + 2t – 1, y = 3t + 5 represents

1) an ellipse 2) a hyperbola 3) a parabola 4) a circle Solution: (3) a parabola Given x = t2 + 2t - 1 & y = 3t + 5 ⇒ x = t2 + 2t + 1 - 2... View Article

Equation of the parabola with its vertex at (1, 1) and focus (3, 1) is

1) (x -1)2 = 8 (y - 1) 2) (y -1)2 = 8 (x - 3) 3) (y -1)2 = 8 (x - 1) 4) (x -3)2 = 8 (y - 1) Solution: (3) (y -1)2 = 8 (x - 1) The vertex of... View Article

A hyperbola, having the transverse axis of length 2 sin θ, is confocal with the ellipse 3x² + y² = 12 then, its equation is

1) x2 cosec2 θ - y2 sec2 θ = 1 2) x2 sec2 θ - y2 cosec2 θ = 1 3) x2 sin2 θ - y2 cos2 θ = 1 4) x2 cos2 θ - y2 sin2 θ = 1 Solution: (1) x2... View Article

If O is the origin and A is a point on the curve y² = 4x. Then, the locus of the mid – point of OA, is

1) x2 = 4y 2) x2 = 2y 3) x2 = 16y 4) y2 = 2x Solution: (4) y2 = 2x Origin O (0, 0) and A (x, 4) If (x1, y1) and (x2, y2) are the two... View Article

A point P moves, so that the difference of its distances from (-ae, 0) and (ae, 0) is 2a. Then, the locus of P is

1) (x2 / a2) - (y2 / [a2 (e2 - 1)]) = 1 2) (x2 / a2) + (y2 / [a2 (1 - e2)]) = 1 3) (x2 / a2) + (y2 / [a2 (1 + e2)]) = 1 4) (x2 / a2) - (y2 /... View Article

For an ellipse with eccentricity 1 / 2 the centre is at the origin. If one directrix is x = 4, then the equation of the ellipse is

1) 3x2 + 4y2 = 1 2) 3x2 + 4y2 = 12 3) 4x2 + 3y2 = 1 4) 4x2 + 3y2 = 12 Solution: (2) 3x2 + 4y2 = 12 The equation of directrix x = 4 which is... View Article

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