The line $y = x + c$ will intersect the circle $x^2 + y^2 = 1$ in two coincident points,if

Two sides of a square are along the lines $x=-5$ and $y=4$. The point of intersection of the diagonals is $(3,-4)$. The point of intersection of the tangents drawn to the circumcircle of the square at the two consecutive vertices lying on $x=-5$ is

When are the tangents drawn from the origin to the circle $x^2 + 2px + y^2 - 2qy + q^2 = 0$ perpendicular to each other?

$A$ tangent line $L$ is drawn at the point $(2, -4)$ on the parabola $y^{2} = 8x$. If the line $L$ is also tangent to the circle $x^{2} + y^{2} = a$,then $a$ is equal to .... .

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

Suppose two tangents $PA$ and $PB$ are drawn to the circle centered at $C(1, 2)$ from the point $P(16, 7)$. If the area of the quadrilateral $PACB$ is $75$ square units,then the radius of the circle is: