For the circle $x^2+y^2-9=0$,find the equation of the chord having $(1,2)$ as its mid-point.

The length of the chord joining the points $(4 \cos \theta, 4 \sin \theta)$ and $(4 \cos (\theta+60^{\circ}), 4 \sin (\theta+60^{\circ}))$ of the circle $x^{2}+y^{2}=16$ is

The equation of the circle passing through $(2,0)$ and $(0,4)$ and having the minimum radius is:

The circle $x^2 + y^2 - 4x - 8y - 5 = 0$ will intersect the line $3x - 4y = m$ in two distinct points,if:

Let $A(3,5)$,$B(-2,-7)$ and $C(\alpha, \beta)$ be three points such that $\angle ACB$ is a right angle and the area of triangle $ABC$ is $\frac{82}{3}$ square units. Then the number of such points $C$ is

The equation of the circle with $(1,1)$ as centre and which cuts a chord of length $4 \sqrt{2}$ units on the line $x+y+1=0$ is