$x-2y-6=0$ 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

If the line $(x + g) \cos \theta + (y + f) \sin \theta = k$ is tangent to the circle $x^2 + y^2 + 2gx + 2fy + c = 0$,then:

Points $P(-3, 2)$,$Q(9, 10)$,and $R(\alpha, 4)$ lie on a circle $C$ with $PR$ as its diameter. The tangents to $C$ at the points $Q$ and $R$ intersect at the point $S$. If $S$ lies on the line $2x - ky = 1$,then $k$ is equal to $.........$.

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

Find the equations of the tangents to the circle $x^2 + y^2 - 6x + 4y = 12$ which are parallel to the line $4x + 3y + 5 = 0$.

If a line,$y=mx+c$ is a tangent to the circle,$(x-3)^{2}+y^{2}=1$ and it is perpendicular to a line $L_{1},$ where $L_{1}$ is the tangent to the circle,$x^{2}+y^{2}=1$ at the point $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right),$ then