If the length of the tangents drawn from the point $(1,2)$ to the circles ${x^2} + {y^2} + x + y – 4 = 0$ and $3{x^2} + 3{y^2} – x – y + k = 0$ be in the ratio $4 : 3$, then $k =$

If $\theta $ is the angle subtended at $P({x_1},{y_1})$ by the circle $S \equiv {x^2} + {y^2} + 2gx + 2fy + c = 0$, then

Let the lines $y+2 x=\sqrt{11}+7 \sqrt{7}$ and $2 y + x =2 \sqrt{11}+6 \sqrt{7}$ be normal to a circle $C:(x-h)^{2}+(y-k)^{2}=r^{2}$. If the line $\sqrt{11} y -3 x =\frac{5 \sqrt{77}}{3}+11$ is tangent to the circle $C$, then the value of $(5 h-8 k)^{2}+5 r^{2}$ is equal to…….

The equations of the tangents to circle $5{x^2} + 5{y^2} = 1$, parallel to line $3x + 4y = 1$ are

The line $ax + by + c = 0$ is a normal to the circle ${x^2} + {y^2} = {r^2}$. The portion of the line $ax + by + c = 0$ intercepted by this circle is of length