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Let A and B be two distinct points on the parabola y² = 4x. If the axis of the parabola touches a circle of radius 2 having AB as its diameter, then the slope of the line joining A and B can be

1) - (1/2) 2) 1/2 3) 1 4) None of these Solution: (3) 1 Let the 2 points be A (at12, 2at1) and B (at22, 2at2) a = 1, mid point of AB = [(t12... View Article

If the line y – √3x + 3 = 0 cuts the parabola y² = x + 2 at A and B, then PA. PB is equal to [where, P = (√3, 0)]

1) [4 (2 - √3)] / 3 2) [4 (√3 + 2)] / 3 3) 4√3 / 3 4) [2 (√3 + 2)] / 3 Solution: (2) [4 (√3 + 2)] / 3 Given P = (√3, 0) The equation of line... View Article

The locus of a point that moves such that the difference of its distance from two fixed points is always a constant, is

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

The eccentricity of the ellipse 25x² + 16y² – 150x – 175 = 0 is

1) 2/5 2) 2/3 3) 4/5 4) 3/4 5) 3/5 Solution: (5) 3/5 25x2 + 16y2 - 150x - 175 = 0 The given equation can be reduced to [x - 3]2 / 16 + [y2... View Article

The eccentricity of the hyperbola (x² / 16) – (y² / 25) = 1 is

1) 3/4 2) 3/5 3) √41/4 4) √41/5 Solution: (3) √41/4 (x2 / 16) - (y2 / 25) = 1 a2 = 16, b2 = 25 e = √(a2 + b2) / a2 = √(16 + 25) / 16 = √41 /... View Article

The distance between the directrices of the hyperbola x = 8 secθ, y = 8 tanθ is

1) 8√2 2) 16√2 3) 4√2 4) 6√2 Solution: (1) 8√2 Given, x = 8 sec θ and y = 8 tan θ => sec θ = x / 8 and tan θ = y / 8 sec2 θ - tan2θ = 1... View Article

If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of its latus rectum is

1) 4√3 2) 4/√3 3) √3 / 4 4) 3 √2 Solution: (4) 3 √2 2ae = 6 and 2a/e = 12 ⇒ ae = 3 and a / e = 6 ⇒ a2 = 18 ⇒ b2 = a2 - a2e2 = 18 - 9 = 9... View Article

An ellipse has OB as semi-minor axis F and F’ its foci and the angle FBF’ is a right angle. Then, the eccentricity of the ellipse is

1) 1/√3 2) 1/4 3) 1/2 4) 1/√2 Solution: (4) 1/√2 Since, F and F’ are foci of an ellipse whose coordinates are (ae, 0) and (- ae, 0)... View Article

The latus rectum of the parabola y² = 4ax, whose focal chord is PSQ, such that SP = 3 and SQ = 2 is given by

1) 24/5 2) 12/5 3) 6/5 4) 1/5 Solution: (1) 24/5 Since 2l = 1 SP + 1SQ (where l is the semi-latus rectum of the parabola) 2l = (1 / 3) + (1... View Article

The focal chord to y² = 16x is tangent to (x – 6)² + y² = 2, then the possible values of the slope of this chord, are

1) {–1, 1} 2) {–2, 2} 3) {–2, –1/2} 4) {2, –1/2} Solution: (1) {–1, 1} The focal chord to y2 = 16x is tangent to (x – 6)2 + y2 = 2. The... View Article

If the eccentricity of a hyperbola is √3, then the eccentricity of its conjugate hyperbola is

1) √2 2) √3 3) √3 / 2 4) 2√3 Solution: (3) √3 / 2 (1 / e2) + (1 / e’2) = 1 It is given that e = √3. (1 / e’2) = 1 - (1 / √3)2 (1 / e’2) = 1... View Article

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