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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

Length of the straight line x – 3y = 1 intercepted by the hyperbola x² – 4y² = 1 is

1) (3/5) √10 2) (6/5) √10 3) (5/3) √10 4) (5/6) √10 Solution: 2) (6/5) √10 Given line is x - 3y = 1 => x = 1 + 3y ..(i) Equation is... View Article

The locus of the equation x² + y² = 0, is

1) a circle 2) a hyperbola 3) a pair of lines 4) at the origin Solution: (4) at the origin x2 + y2 = 0 x2 = - y2 A perfect square can never... View Article

If t is a parameter then x = a(t + 1/t) and y = b(t – 1/t) represents

1) an ellipse 2) a circle 3) a pair of straight lines 4) hyperbola Solution: (4) hyperbola x = a(t + 1/t) and y = b(t - 1/t) (x / a) = (t +... View Article

The equation of the ellipse having vertices at (± 5, 0) and foci (± 4, 0) is

1) (x2 / 25) + (y2 / 16) = 1 2) 9x2 + 25y2 = 225 3) (x2 / 9) + (y2 / 25) = 1 4) 4x2 + 5y2 = 20 Solution: (2) 9x2 + 25y2 = 225 c = 4, a = 5... View Article

The two parabolas x² = 4y and y² = 4x meet in two distinct points. One of these is the origin and the other is

1) (2, 2) 2) (4, -4) 3) (4, 4) 4) (-2, 2) Solution: (3) (4, 4) x2 = 4y and y2 = 4x --- (i) are the given equations of parabola. (x2 / 4)2 =... View Article

The locus of the mid – point of the line joining focus and any point on the parabola y² = 4ax is a parabola with the equation of directrix as

1) x + a = 0 2) 2x + a = 0 3) x = 0 4) x = a / 2 Solution: (3) x = 0 If a point P (h, k) be the midpoint of the line segment joining the... View Article

The equation (x² / (2 – λ)) – y² / (λ – 5) – 1 = 0 represents an ellipse, if

1) λ > 5 2) λ < 2 3) 2 < λ < 5 4) 2 > λ > 5 Solution: (2) λ < 2 (x2 / (2 - λ)) - y2 / (λ - 5) - 1 = 0 x2 / [2 - λ] + y2... View Article

The equation of the parabola whose focus is (3, – 4) and directrix 6x – 7y + 5 = 0, is

1) (7x + 6y)2 - 570x + 750y + 2100 = 0 2) (7x + 6y)2 + 570x - 750y + 2100 = 0 3) (7x - 6y)2 - 570x + 750y + 2100 = 0 4) (7x - 6y)2 + 570x -... View Article

x = 4 (1 + cos θ) and y = 3 (1 + sinθ) are the parametric equations of

1) [(x - 3)2 / 9] + [(y - 4)2 / 16] = 1 2) [(x + 4)2 / 16] + [(y + 3)2 / 9] = 1 3) [(x - 4)2 / 16] - [(y - 3)2 / 9] = 1 4) [(x - 4)2 / 16] +... View Article

Locus of the point which divides double ordinate of the ellipse (x² / a²) + (y² / b²) = 1 in the ratio 1 : 2 internally is

1) (x2 / a2) - (9y2 / b2) = 1 / 9 2) (x2 / a2) + (9y2 / b2) = 1 3) (9x2 / a2) + (9y2 / b2) = 1 4) None of these Solution: (2) (x2 / a2) +... View Article

A parabola is drawn with its focus at (3, 4) and vertex at the focus of the parabola y² – 12x – 4y + 4 = 0. The equation of the parabola is

1) y2 - 8x - 6y + 25 = 0 2) y2 - 6x + 8y - 25 = 0 3) x2 - 6x - 8y + 25 = 0 4) x2 + 6x - 8y - 25 = 0 Solution: (3) x2 - 6x - 8y + 25 = 0 y2 -... View Article

The equation of the ellipse whose distance between the foci is equal to 8 and distance between the directrix is 18, is

1) 5x2 - 9y2 = 180 2) 9x2 + 5y2 = 180 3) x2 + 9y2 = 180 4) 5x2 + 9y2 = 180 Solution: (4) 5x2 + 9y2 = 180 (x2 / a2) + (y2 / b2) = 1 is the... View Article

Consider the two curves C₁ : y² = 4x and C₂ : x² + y² – 6x + 1 = 0, then

1) C1 and C2 touch each other at only one point 2) C1 and C2 touch each other at exactly two points 3) C1 and C2 intersect (but do not touch )... View Article

The circle x² + y² = a² intersects the hyperbola xy = c² in four points (x_i, y_i), for i = 1, 2, 3 and 4, then y₁ + y₂ + y₃ + y₄ equals

1) 0 2) c 3) a 4) c4 Solution: (1) 0 x2 + y2 = a2 --- (1) and xy = c2 ---- (ii) x2 + c4 / x2 = a2 x4 - a2x2 + c4 = 0 ---- (iii) x1, x2, x3, x4... View Article

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