The length of the tangent from the point $(4, 5)$ to the circle $x^2 + y^2 + 2x - 6y - 6 = 0$ is

The combined equation of the tangent and normal to the curve $xy = 100$ at the point $(5, 20)$ is . . . . . . .

The equation of the normal to the circle $x^2+y^2=16$ at the point $\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)$ is

If the normals of the parabola $y^2 = 4x$ drawn at the end points of its latus rectum are tangents to the circle $(x - 3)^2 + (y + 2)^2 = r^2$,then the value of $r^2$ is

The length of the tangent drawn from the mid-point of the line joining the origin and the point $(4, -4)$ to the circle $2x^2 + 2y^2 - y = 0$ 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: