The slope of the normal to the circle $x^2+y^2+2gx+2fy+c=0$ at $(x_1, y_1)$ is

Let $AB$ be a chord of length $12$ of the circle $(x-2)^{2}+(y+1)^{2}=\frac{169}{4}$. If tangents drawn to the circle at points $A$ and $B$ intersect at the point $P$,then five times the distance of point $P$ from chord $AB$ is equal to $.......$

One end of the diameter of the circle $x^2+y^2-6x-5y-1=0$ is $(-1,3)$. Find the equation of the tangent at the other end of the diameter.

The tangent and the normal lines at the point $(\sqrt{3}, 1)$ to the circle $x^2 + y^2 = 4$ and the $x$-axis form a triangle. The area of this triangle (in square units) is

$A$ triangle is formed by the tangents at the point $(2,2)$ on the curves $y^2=2x$ and $x^2+y^2=4x$,and the line $x+y+2=0$. If $r$ is the radius of its circumcircle,then $r^2$ is equal to $........$.

If $a > 2b > 0$,then the positive value of $m$ for which $y = mx - b\sqrt{1 + m^2}$ is a common tangent to $x^2 + y^2 = b^2$ and $(x - a)^2 + y^2 = b^2$ is: