If the line $x = 7$ touches the circle ${x^2} + {y^2} - 4x - 6y - 12 = 0$,then the coordinates of the point of contact are:

Consider a circle $(x-\alpha)^2+(y-\beta)^2=50$,where $\alpha, \beta > 0$. If the circle touches the line $y+x=0$ at the point $P$,whose distance from the origin is $4 \sqrt{2}$,then $(\alpha+\beta)^2$ is equal to................

If one common tangent of the two circles $x^2 + y^2 = 4$ and $x^2 + (y - 3)^2 = \lambda, \lambda > 0$ passes through the point $(\sqrt{3}, 1)$,then the possible value of $\lambda$ is

The equation of a common tangent to the circle $x^2 + y^2 - 6x = 0$ and the parabola $y^2 = 4x$ is:

The equations of the tangents to the circle $x^{2}+y^{2}=13$ at the points whose abscissa is $2$ are:

If the pair of tangents drawn to the circle $x^2+y^2=a^2$ from the point $(10,4)$ are perpendicular,then $a=$