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

Point $M$ moves along the circle $(x - 4)^2 + (y - 8)^2 = 20$. It then breaks away from the circle and moves along a tangent to the circle,passing through the point $(-2, 0)$ on the $x$-axis. The coordinates of the point on the circle at which the moving point broke away can be:

$x-2y-6=0$ is a normal to the circle $x^2+y^2+2gx+2fy-8=0$. If the line $y=2$ touches this circle,then the radius of the circle can be

The equation of the circle passing through the point $(-1, -3)$ and touching the line $4x + 3y - 12 = 0$ at the point $(3, 0)$ is

If $3x + y + k = 0$ is a tangent to the circle $x^{2} + y^{2} = 10$,the values of $k$ are

The circle $x=5 \cos \theta, y=5 \sin \theta$ is bounded by the rectangle formed by the lines $x \pm 6=0$ and $y \pm 6=0$. The area of the triangle that lies inside the rectangle,which is formed by the tangent at $P\left(\frac{2 \pi}{3}\right)$ to the circle with two of the above given lines,is