The line $ax + by + c = 0$ is a normal to the circle $x^2 + y^2 = r^2$. The length of the intercept made by the circle on the line $ax + by + c = 0$ is:

The equation of the normal to the curve $x^{2}+y^{2}=r^{2}$ at the point $P(r \cos \theta, r \sin \theta)$ is:

The tangent to the circle $C_1 : x^2 + y^2 - 2x - 1 = 0$ at the point $(2, 1)$ cuts off a chord of length $4$ from a circle $C_2$ whose centre is $(3, -2)$. The radius of $C_2$ is

If $PA$ and $PB$ are the tangents drawn from the point $P(1,1)$ to the circle $x^2+y^2+gx+gy-2=0$ with $C$ as the centre,then the area (in sq. units) of the quadrilateral $PACB$ is

If $x-2y=0$ is a tangent drawn at a point $P$ on the circle $x^2+y^2-6x+2y+c=0$,then the distance of the point $(6,3)$ from $P$ is

Let the centre of a circle $C$ be $(\alpha, \beta)$ and its radius $r < 8$. Let $3x + 4y = 24$ and $3x - 4y = 32$ be two tangents and $4x + 3y = 1$ be a normal to $C$. Then $(\alpha - \beta + r)$ is equal to $........$.