Find the condition that the straight line cx – by + b² = 0 may touch the circle x² + y² = ax + by and find the point of contact.

Find whether the straight line  touches the circle x² + y² – 2x – 2y + 1 = 0.

Find the condition that the line 3x + 4y = k may touch the circle x² + y² = 10x.

Find the value of p so that the straight line xcosα + ysinα – p = 0 may touch the circle x² + y² – 2axcosα – 2bysinα = a²sin²α

Find the condition that the straight line Ax + By + C = 0 may touch the circle(x – a)² + (y – b)² = c²

Find the equations of tangents to the circle x² + y² = a² which is parallel to the straight line y = mx + c.

Find the equations of tangents to the circle x² + y² = a² which is perpendicular to the straight line y = mx + c

Find the equations of tangents to the circle x² + y² = a² which makes with axes a triangle whose area is a²

Find the length of the chord joining the points in which the straight line  meets the circle x² + y² = r²

Find the equations to the circle which has its centre at the point (3, 4) and touches the straight line 5x + 12y = 1.

Find the equation to the circle which touches the axes of coordinates and also the line  

Find the equation to the circle which has its centre at the point (1, −3) and touches the straight line 2x – y – 4 = 0.

Find the general equation of a circle referred to two perpendicular tangents as axes.

Find the polar of the points (−2, 3) with respect to the circle x² + y² – 4x – 6y + 5 = 0.

a) x = 0

b) y = 0

c) x = y

d) x + y = 1

Find the polar of the point (a, − b) with respect to the circle x² + y² + 2ax – 2by + a² – b² = 0

a) ay – bx = a²

b) by – ax = a²

c) by – ax = b²

d) ay – bx = b²

Find the pole of the straight line 2x + y + 12 with respect to the circle x² + y² – 4x + 3y – 1 = 0.

a) (−1, 2)

b) (1, 2)

c) (1, −2)

d) (−1, −2)

Find the pole of the straight line ax + by +3a² + 3 b² = 0 with respect to the circle x² + y² + 2ax + 2by = a² + b² 

a) (a, b)

b) (−a, −b)

c) (2a, 2b)

d) (−2a, −2b)

Find the condition that the chord of the contact of the tangents from the point (x’, y’) to the circle x² + y² = a² subtend a right angle at the centre.

a)

b)

c)

d)

Prove that the distances of two points P and Q, each from the polar of the other with respect to a circle, are to each other as the distance of the points from the centre of the circle.

Prove that the polar of a given point with respect to any one of the circle x² + y² − 2kx + c² = 0 where k is a variable, always passes through a fixed point, whatever be the value of k.

Find the length of tangents drawn to the circle x² + y² +2bx – 3b² = 0 from the point (a + b, a – b).

a)

b)

c) a + b

d)

The distance from the origin of the centre of three circles x² + y² − 2λx = c² (where c is a constant and λ a variable) are in geometric progression. Then the lengths of the tangents drawn to them from any point on the circle x² + y² = c² are also in

a) Geometric Progression.

b) Arithmetic Progression

c) Harmonic Progression

d) None of these

Find the coordinates of the centre of the circle
r = Acosθ + Bsinθ

a) (A, B)

b)

c) (2A, 2B)

d)