$A$ circle with centre $(a, b)$ passes through the origin. The equation of the tangent to the circle at the origin is

If the straight line $y = mx + c$ touches the circle $x^2 + y^2 - 2x - 4y + 3 = 0$ at the point $(2, 3)$,then $c =$

If the tangents $x+y+k=0$ and $x+ay+b=0$ drawn to the circle $S \equiv x^2+y^2+2x-2y+1=0$ are perpendicular to each other and $k, b$ are both greater than $1$,then $b-k=$

If the length of the tangent drawn from the point $(5, 3)$ to the circle $x^2 + y^2 + 2x + ky + 17 = 0$ is $7$,then the value of $k$ is:

When is the line $lx + my + n = 0$ a tangent to the circle $x^2 + y^2 = r^2$?

$A$ line is a common tangent to the circle $(x-3)^{2}+y^{2}=9$ and the parabola $y^{2}=4x$. If the two points of contact $(a, b)$ and $(c, d)$ are distinct and lie in the first quadrant,then $2(a+c)$ is equal to ........ .