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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

Focus of hyperbola is (± 3, 0) and equation of tangent is 2x + y – 4 = 0, find the equation of hyperbola is

1) 4x2 - 5y2 = 20 2) 5x2 - 4y2 = 20 3) 4x2 - 5y2 = 1 4) 5x2 - 4y2 = 1 Solution: (1) 4x2 - 5y2 = 20 Let the hyperbola be x2 / a2 - y2 / b2 =... View Article

The ellipse x² + 4y² = 4 is inscribed in a rectangle aligned with the coordinate axes, which is turn in inscribed in another ellipse that passes through the point (4, 0). Then, the equation of the ellipse is

1) x2 + 12y2 = 16 2) 4x2 + 48y2 = 48 3) 4x2 + 64y2 = 48 4) x2 + 16y2 = 16 Solution: (1) x2 + 12y2 = 16 The given ellipse is x2 / 4 + y2 / 1... View Article

The circle x² + y² – 8x = 0 and hyperbola x²/ 9 – y² / 4 = 1 intersect at the points A and B. Equation of the circle with AB as its diameter is

1) x2 + y2 - 12x + 24 = 0 2) x2 + y2 + 12x + 24 = 0 3) x2 + y2 - 24x - 12 = 0 4) x2 + y2 - 24x - 12 = 0 Solution: (1) x2 + y2 - 12x + 24 = 0... View Article

The circle x² + y² – 8x = 0 and hyperbola x² / 9 – y² / 4 = 1 intersect at the points A and B. Equation of a common tangent with positive slope to the circle as well as to the hyperbola is

1) 2x - √5y - 20 = 0 2) 2x - √5y + 4 = 0 3) 3x - 4y + 8 = 0 4) 4x - 3y + 4 = 0 Solution: (2) 2x - √5y + 4 = 0 Equation of tangent to circle... View Article

The parabola y² = 4x and the circle x² + y² – 6x + 1 = 0 will

1) intersect at exactly one point 2) touch each other at two distinct points 3) touch each other at exactly one point 4) intersect at two... View Article

The distance between the vertex of the parabola y = x² – 4x + 3 and the centre of the circle x² = 9 – (y – 3)² is

1) 2√3 units 2) 3√2 units 3) 2√2 units 4) √2 units 5) 2√5 units Solution: (5) 2√5 units y = x2 - 4x + 3 = x2 - 4x + 4 - 1 = (x - 2)2 - 1 y... View Article

Equation of the ellipse whose axes are the axes of coordinates and which passes through the point (-3, 1) and has eccentricity √(2 / 5) is

1) 5x2 + 3y2 - 48 = 0 2) 3x2 + 5y2 - 15 = 0 3) 5x2 + 3y2 - 32 = 0 4) 3x2 + 5y2 - 32 = 0 Solution: (4) 3x2 + 5y2 - 32 = 0 It is given that... View Article

If the eccentricity of the hyperbola x² / a² – y² / b² = 1 is reciprocal to that of the ellipse x² + 4y² = 4. If the hyperbola passes through the focus of the ellipse, then

1) the equation of the hyperbola is x2 / 3 - y2 / 2 = 1 2) a focus of the hyperbola is (2, 0) 3) the eccentricity of the hyperbola is √(5/3)... View Article

Let P and Q are the points on the parabola y² = 4x, if so that the line segment PQ subtends a right angle at the vertex. If PQ intersects the axis of the parabola at R, then the distance of the vertex from R is

1) 1 2) 2 3) 4 4) 6 Solution: (3) 4 (t12, 2t2) and (t22, 2t2) respectively and let the coordinates of R be (x, 0). x = [t12 + t22] / 2, y =... View Article

If the foci of the ellipse x² / 9 + y² = 1 subtend right angle at a point P. Then, the locus of P is

1) x2 + y2 = 1 2) x2 + y2 = 2 3) x2 + y2 = 4 4) x2 + y2 = 8 Solution: (4) x2 + y2 = 8 x2 / 9 + y2 = 1 a = 3, b = 1 e2 = 1 - b2 / a2 = 1 - (1... View Article

The line x = 2y intersects the ellipse x² / 4 + y² = 1 at the points P and Q. The equation of the circle with PQ as diameter is

1) x2 + y2 = 1/2 2) x2 + y2 = 1 3) x2 + y2 = 2 4) x2 + y2 = 5/2 Solution: (4) x2 + y2 = 5/2 x2 / 4 + y2 = 1 4y2 / 4 + y2 = 1 2y2 = 1 y2 = 1... View Article

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