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The eccentricity of the hyperbola 9x² – 16y² – 18x – 64y – 199 = 0 is

1) 16/9 2) 5/4 3) 25/16 4) 0 Solution: (2) 5 / 4 9x2 - 16y2 - 18x - 64y - 199 = 0 9 [(x - 1)2 - 1] - 16 [(y + 2)2 - 4] = 199 9 (x - 1)2 - 9... View Article

The eccentricity of the hyperbola conjugate to x² – 3y² = 2x + 8 is

1) 2/√3 2) √3 3) 2 4) None of these Solution: (3) 2 x2 - 3y2 = 2x + 8 The equation reduces to (x - 1)2 / 9 - (y2 / 3) = 1. Its conjugate... View Article

The equation of the directrix of the parabola x² + 8y – 2x = 7 is

1) y = 3 2) y = -3 3) y = 2 4) y = 0 Solution: (1) y = 3 x2 + 8y - 2x = 7 ⇒ x2 - 2x + 8y - 7 = 0 ⇒ x2 - 2x + 1 + 8y - 7 -1 = 0 ⇒ (x -1)2 +... View Article

Suppose S and S’ are foci of the ellipse (x² / 25) + (y² / 16) = 1. If P is a variable point on the ellipse and △ is the area of the △PSS’ then the maximum value of △ is

1) 8 2) 12 3) 16 4) 20 Solution: (2) 12 (x2 / 25) + (y2 / 16) = 1 as major axis since it is x-axis. a > b Foci = (± ae, 0) e = √1 -... View Article

The ends of the latus rectum of the conic x² + 10x – 16y + 25 = 0 are

1) (3, - 4), (13, 4) 2) (-3, -4), (13, -4) 3) (3, 4), (-13, 4) 4) (5, -8), (-5, 8) Solution: (3) (3, 4), (-13, 4) x2 + 10x - 16y + 25 = 0 ⇒... View Article

The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y² = 4ax is another parabola with directrix

1) x = -a 2) x = -a/2 3) x = 0 4) x = a/2 Solution: (3) x = 0 Let P (h, k) be the midpoint of the line segment joining the focus (a, 0) and... View Article

If a > 2b > 0 then the positive value of m for which y = mx – b √1 + m² is a common tangent to x² + y² = b² and (x – a)² + y² = b² is

1) 2b / √a2 - 4b2 2) √a2 - 4b2 / 2b 3) 2b / (a - 2b) 4) b / (a - 2b) Solution: (1) 2b / √a2 - 4b2 y = mx - b √1 + m2 touches both the... View Article

Coordinates of the foci of the ellipse 5x² + 9y² + 10x – 36y – 4 = 0, are

1) (1, 2) and (-3, 2) 2) (2, 1) and (-3, 2) 3) (1, 2) and (3, 2) 4) None of these Solution: (1) (1, 2) and (-3, 2) 5x2 + 9y2 + 10x - 36y - 4... View Article

The equation of the directrix of the parabola y² + 4y + 4x + 2 = 0 is

1) x = -1 2) x = 1 3) x = -3/2 4) x = 3/2 Solution: (4) x = 3/2 y2 + 4y + 4x + 2 = 0 (y + 2)2 = - 4 [x - (1 / 2)] Let y + 2 = Y and x - (1 /... View Article

The equation of the common tangent touching the circle (x – 3)² + y² = 9 and the parabola y² = 4x above the x-axis is

1) √3y = 3x + 1 2) √3y = - (x + 3) 3) √3y = x + 3 4) √3y = - (3x + 1) Solution: (3) √3y = x + 3 The tangent to y2 = 4x is y = mx + 1. It... View Article

If for the ellipse (x² / a²) + (y² / b²) = 1, the y-axis is the minor axis and the length of the latus rectum is one half of the length of its minor axis, then its eccentricity is

1) 1/√2 2) 1/2 3) √3/2 4) 3/4 5) 3/5 Solution: (3) √3/2 Given the equation of the ellipse is (x2 / a2) + (y2 / b2) = 1. By the question 2b2... View Article

The eccentricity of the conic [(x + 2)² / 7] + (y – 1)² = 14 is

1) √7 / 8 2) √6 / 17 3) √3 / 2 4) √6 / 11 5) √6 / 7 Solution: (5) √6 / 7 a2 = 7 * 14 b2 = 14 e = √1 - (b2 / a2) = √1 - (1 / 7) = √6 / 7... View Article

If P is any point on the ellipse (x² / 36) + (y² / 16) = 1 and S and S’ are the foci, then PS + PS’ =

1) 4 2) 8 3) 10 4) 12 Solution: (4) 12 Given, the ellipse is (x2 / 36) + (y2 / 16) = 1 Here a2 = 36, b2 = 16 Since, a > b, so the sum of... View Article

The equation of the latusrectum of the parabola x² + 4x + 2y = 0, is equal to

1) 2y + 3 = 0 2) 3y = 2 3) 2y = 3 4) 3y + 2 = 0 Solution: (3) 2y = 3 (x + 2) 2 = -2 (y - 2) The equation of latus rectum is y - 2 = - 1 / 2... View Article

The equation of a directrix of the ellipse (x² / 16) + (y² / 25) = 1 is

1) 3y = ± 5 2) y = ± 5 3) 3y = ± 25 4) y = ± 3 Solution: (3) 3y = ± 25 It is given that a = 4, b = 5 and e = √1 - (16 / 25) = 3 / 5 The... View Article

The foci of the conic section 25x² + 16y² – 150x = 175 are

1) (0, ± 3) 2) (0, ± 2) 3) (3, ± 3) 4) (0, ± 1) Solution: (3) (3, ± 3) 25x2 + 16y2 - 150x = 175 25 (x2 - 6x + 9 - 9) + 16y2 = 175 25 (x -... View Article

For the hyperbola, (x² / cos² ɑ) – (y² / sin² ɑ) = 1, which of the following remains constant when ɑ varies?

1) Eccentricity 2) Directrix 3) Abscissae of vertices 4) Abscissae of foci Solution: (4) Abscissae of foci ∵ b2 = a2 (e2 - 1) => sin2α =... View Article

Statement I the curve y = – (x² / 2) + x + 1 is symmetric with respect to the line x = 1. Statement II A parabola is symmetric about its axis

1) Statement I is correct, Statement II is correct; Statement II is a correct explanation for Statement I 2) Statement I is correct, Statement... View Article

Equation of the latusrectum of the ellipse 9x² + 4y² – 18x – 8y – 23 = 0

1) y = ±√5 2) y = -√5 3) y = 1 ± √5 4) x = -1± √5 Solution: (3) y = 1 ± √5 9x2 + 4y2 - 18x - 8y - 23 = 0 = (3x - 3)2 + (2y - 2)2 - 13 - 23 =... View Article

An equilateral triangle is inscribed in the parabola y² = 4ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.

1) 8a√3 2) 4a√3 3) 2a√3 4) a√3 Solution: (1) 8a√3 Let the 2 vertices of the triangle be Q and R. The points Q and R will have the same... View Article