The sum of the slopes of the lines given by x² – 2cxy – 7y² = 0 is four times their product. Then the value of c is given by

a) 1

b) – 1

c) 2

d) – 2

If one of the lines given by 6x² – xy + 4cy² = 0 is 3x + 4y = 0 then c equals

a) 1

b) – 1

c) 3

d) – 3

The lines 2x + 3y + 1 = 0 and 3x – y – 4 = 0 tie along diameter of a circle of circumference 10π. Then the equation of the circle is

a) x² + y² – 2x + 2y – 23 = 0

b) x² + y² – 2x – 2y – 23 = 0

c) x² + y² + 2x + 2y – 23 = 0

d) x² + y² + 2x – 2y – 23 = 0

The intercept on the line y = x by the circle x² + y² – 2x = 0 is AB. The equation of the circle with AB as diameter is

a) x² + y² – x – y = 0

b) x² + y² – x +  y = 0

c) x² + y² + x + y = 0

d) x² + y² + x – y = 0

If a ≠ 0 and the line 2bx + 3cy + 4d = 0 passes through the point of intersection of the parabolas y² = 4ax and x² = 4ay then

a) d² + (2b + 3c)² = 0

b) d² + (3b + 2c)² = 0

c) d² + (2b – 3a)² = 0

d) d² + (3b – 2a)² = 0

The eccentricity of an ellipse, with its centre as origin, is . If one of the directrix is x = 4 then the equation of the ellipse is

a) 3x² + 4y² = 1

b) 3x² + 4y² = 12

c) 4x² + 3y² = 12

d) 4x² + 3y² = 1

If x₁, x₂, x₃ and y₁, y₂ and y₃ are both in Geometric Progression with the same common ratio then (x₁, y₁), (x₂, y₂), (x₃, y₃)

a) Lie on the same straight line

b) Lie on an ellipse

c) Lie on a circle

d) Are vertices of a triangle

If the equation of the locus of a point equidistant from the points (a₁, b₁) and (a₂, b₂) is (a₁ – a₂) x + (b₁ – b₂) y + c = 0, then the value of c is

a)

b)

c)

d)

The locus of centroid of the triangle whose vertices are (acost, asint), (bsint, − bcost) and (1, 0), where t is a parameter is

a) (3x – 1)² + (3y)² = a² – b²

b) (3x – 1)² + (3y)² = a² + b²

c) (3x + 1)² + (3y)² = a² + b²

d) (3x + 1)² + (3y)² = a² – b²

The circles (x – 1)² + (y – 3)² = r² and x² + y² – 8x + 2y + 8 = 0 intersect in two distinct points, then

a) 2 < r < 8

b) r < 2

c) r = 2

d) r > 2

The normal at any point  on the parabola y² = 4bx meets the parabola again at the point , then

a)

b)

c)

d)

The foci of the ellipse  and the hyperbola  coincide, then the value of b² is

a) 1

b) 5

c) 7

d) 9

A triangle with vertices (4, 0), (−1, −1), (3, 5) is

a) Isosceles and right angled

b) Isosceles but not right angled

c) Right angled but not isosceles

d) Neither isosceles nor right angled

The incentre of the triangle with vertices  is

a)

b)

c)

d)

Let the radii of the circle passing through the foci of the ellipse  and having its centre as (0, 3) is

a) 4 units

b) 3 uits

c)  units

d)  units

Three straight lines
2x + 11y – 5 = 0
24x + 7y – 20 = 0
4x – 3y – 2 = 0

a) Form a triangle

b) Are only concurrent

c) Are concurrent with one line bisecting the angle between the other two

d) None of the above

The straight line through the point (2, 2) intersects the lines and  at the points A and B. the equation to the line AB so that ΔOAB is equilateral, is

a) x – 2 = 0

b) y – 2 = 0

c) x + y – 4 = 0

d) None of these

The greatest distance of the point P (10, 7) from the circle x² + y² – 4x – 2y – 20 = 0 is

a) 10 units

b) 15 units

c) 5 units

d) None of these

The equation of the ellipse whose foci are (±2, 0) and eccentricity is , is

a)

b)

c)

d) None of these

If the lines ax + by + c = 0 is tangent to the curve xy = 4 then

a) a > 0, b > 0

b) a > 0, b < 0

c) a < 0, b > 0

d) None of these

The image of the point (−8, 12) with respect to the line mirror 4x + 7y + 13 = 0 is

a) (−16, 2)

b) (16, −2)

c) (16, 2)

d) (−16, −2)

A ray of light along  gets reflected upon reaching X-axis, the equation of the reflected ray is

a) y = x + 1

b)

c)

d) None of these

The portion of a tangent to a parabola y² = 4ax cut off between the directrix and the curve subtends an angle θ at the focus where θ is equal to

a)

b)

c)

d) None of these