If the line $x+3y=0$ is the tangent at $(0,0)$ to the circle of radius $1$,then the centre of one such circle is

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

Let a circle $C$ of radius $5$ lie below the $x$-axis. The line $L_{1}: 4x + 3y + 2 = 0$ passes through the centre $P$ of the circle $C$ and intersects the line $L_{2}: 3x - 4y - 11 = 0$ at $Q$. The line $L_{2}$ touches $C$ at the point $Q$. Then the distance of $P$ from the line $5x - 12y + 51 = 0$ is

The angle between the tangents drawn from the origin to the circle $x^2+y^2+4x-6y+4=0$ is

If the tangent at $(1,7)$ to the curve $x^2=y-6$ touches the circle $x^2+y^2+16x+12y+C=0$,then $C=$

The focal chord of the parabola $(y - 2)^2 = 16(x - 1)$ is a tangent to the circle $x^2 + y^2 - 14x - 4y + 51 = 0$. Then the slope of the focal chord can be: