If the pair of tangents drawn to the circle $x^2+y^2=a^2$ from the point $(10,4)$ are perpendicular,then $a=$

The line $x = y$ touches a circle at the point $(1, 1)$. If the circle also passes through the point $(1, -3)$,then its radius is

The equation of the circle having the lines $y^2 - 2y + 4x - 2xy = 0$ as its normals and passing through the point $(2, 1)$ is:

The angle between the tangents drawn from the point $(2, 2)$ to the circle $x^2 + y^2 + 4x + 4y + c = 0$ is $\cos^{-1}\left(\frac{7}{16}\right)$. If two such circles exist,then the sum of the values of $c$ is:

If the acute angle between the pair of tangents drawn from the origin to the circle $x^2+y^2-4x-8y+4=0$ is $\alpha$,then $\tan \alpha=$

The equation of the normal at $(1, 1)$ to the circle $x^2 + y^2 - x - 3y - 4 = 0$ is