The angle between the pair of tangents drawn from $(1,3)$ to the circle $x^2+y^2-2x+4y-11=0$ is

Let the tangent to the circle $x^{2}+y^{2}=25$ at the point $R(3,4)$ meet the $x$-axis and $y$-axis at points $P$ and $Q$,respectively. If $r$ is the radius of the circle passing through the origin $O$ and having its centre at the incentre of the triangle $OPQ$,then $r^{2}$ is equal to

The normal at a point $\theta$ to the curve $x=a(1+\cos \theta), y=a \sin \theta$ always passes through the fixed point

The equations of the tangents to the circle $x^2 + y^2 = 50$ at the points where the line $x + 7 = 0$ meets it,are

$A$ line is a common tangent to the circle $(x-3)^{2}+y^{2}=9$ and the parabola $y^{2}=4x$. If the two points of contact $(a, b)$ and $(c, d)$ are distinct and lie in the first quadrant,then $2(a+c)$ is equal to ........ .

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: