The equation of tangents to the circle $x^2+y^2=4$ which are parallel to $x+2y+3=0$ are

Let the lines $y+2x=\sqrt{11}+7\sqrt{7}$ and $2y+x=2\sqrt{11}+6\sqrt{7}$ be normal to a circle $C:(x-h)^{2}+(y-k)^{2}=r^{2}$. If the line $\sqrt{11}y-3x=\frac{5\sqrt{77}}{3}+11$ is tangent to the circle $C$,then the value of $(5h-8k)^{2}+5r^{2}$ is equal to.......

The equation of the normal at the point $(4, -1)$ of the circle $x^2 + y^2 - 40x + 10y = 153$ is

The equation of the tangent to the circle at the point $(1, -1)$,whose center is the point of intersection of the straight lines $x - y = 1$ and $2x + y = 3$,is:

The straight line $x+2y=1$ cuts the $X$-axis at $A$ and the $Y$-axis at $B$. $A$ circle is drawn through $A, B$ and the origin $O(0,0)$. The sum of the perpendicular distances from $A$ and $B$ to the tangent drawn at the origin to the circle $S$ is:

$A$ tangent line $L$ is drawn at the point $(2, -4)$ on the parabola $y^{2} = 8x$. If the line $L$ is also tangent to the circle $x^{2} + y^{2} = a$,then $a$ is equal to .... .