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If b and c are the lengths of the segments of any focal chord of a parabola y² = 4ax, then the length of the latus rectum is

1) bc / [b + c] 2) √bc 3) [b + c] / 2 4) [2bc / (b + c)] Solution: (4) [2bc / (b + c)] It is given that b and c are lengths of the segments... View Article

For the ellipse 25x² + 9y² – 150x – 90y + 225 = 0, the eccentricity is equal to

1) 2/5 2) 3/5 3) √15 / 24 4) 4/5 Solution: (4) ⅘ 25x2 + 9y2 - 150x - 90y + 225 = 0 25 (x2 - 6x) + 9 (y2 - 10y) + 225 = 0 25 [x2 - 6x + 9] +... View Article

The area of the quadrilateral formed by the tangents at the endpoints of the latus rectum to the ellipse x² / 9 + (y² / 5) = 1, is

1) 27/4 sq. units 2) 9 sq. units 3) 27/2 sq. units 4) 27 sq. units Solution: (4) 27 sq. units The equation of the tangent at L (ae, b2 / a)... View Article

The eccentricity of the conic 4x² + 16y² – 24x – 32y = 1 is

1) 1/2 2) √3 3) √3/2 4) √3/4 Solution: (3) √3/2 4x2 + 16y2 - 24x - 32y = 1 4 [(x2 - 6x + 9) - 36] + 16 [[y2 - 2y + 1] - 16] = 1 4 (x - 3)2 +... View Article

The equation of the common tangent to the curves y² = 8x and xy = -1 is

1) 3y = 9x + 2 2) y = 2x + 1 3) 2y = x + 8 4) y = x + 2 Solution: (4) y = x + 2 The tangent to the curve y2 = 8x is y = mx + (2 / m). Hence... View Article

If the eccentricity of the ellipse (x² / 4) + (y² / 3) = 1 and the hyperbola (x² / 64) – (y² / b²) = 1 are reciprocals of each other, then b² =

1) 192 2) 64 3) 16 4) 32 5) 128 Solution: (1) 192 (x2 / 4) + (y2 / 3) = 1 e = √1 - (b2 / a2) = √1 - (3 / 4) = 1 / 2 Eccentricity of the... View Article

The eccentricity of the hyperbola in the standard form (x² / a²) + (y² / b²) = 1, passing through (3, 0) and (3√2, 2) is

1) 13/3 2) √13 3) √3 4) √13/3 5) 5/3 Solution: (4) √13/3 (x2 / a2) + (y2 / b2) = 1 It passes through the points (3, 0) and (3√2 , 2). a2 = 9... View Article

The locus of the point which moves such that the ratio of its distance from two fixed points in the plane is always a constant K (< 1) is

1) a hyperbola 2) an ellipse 3) a straight line 4) a circle Solution: (2) an ellipse The locus of the point which moves such that the ratio... View Article

The ellipse (x² / 25) + (y² / 16) = 1 and the hyperbola (x² / 25) – (y² / 16) = 1 have in common

1) centre only 2) centre, foci and directrices 3) centre, foci and vertices 4) centre and vertices Solution: (4) centre and vertices... View Article

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

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