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

The equation of tangent to the circle ${x^2} + {y^2} = {a^2}$ parallel to $y = mx + c$ is

The area of triangle formed by the tangent, normal drawn at $(1,\sqrt 3 )$ to the circle ${x^2} + {y^2} = 4$ and positive $x$-axis, is

The equations of the tangents to the circle ${x^2} + {y^2} = 13$ at the points whose abscissa is $2$, are

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

Let the normals at all the points on a given curve pass through a fixed point $(a, b) .$ If the curve passes through $(3,-3)$ and $(4,-2 \sqrt{2}),$ and given that $a-2 \sqrt{2} b=3,$ then $\left(a^{2}+b^{2}+a b\right)$ is equal to ..... .