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 equation to the circle whose centre is at the point (α, β) and which passes through the origin and prove that the equation of the tangent at the origin is αx + βy = 0.

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