Let a circle $S = 0$ touch both the circles $x^2 + y^2 = 400$ and $x^2 + y^2 - 10x - 24y + 120 = 0$ externally and also touch the $x$-axis. The radius of the circle $S = 0$ is

The range of values of $a$ such that the angle $\theta$ between the pair of tangents drawn from the point $(a, 0)$ to the circle $x^2 + y^2 = 1$ satisfies $\frac{\pi}{2} < \theta < \pi$ is :

Let a common tangent to the curves $y^2=4x$ and $(x-4)^2+y^2=16$ touch the curves at the points $P$ and $Q$. Then $(PQ)^2$ is equal to $..........$.

$A$ pair of tangents are drawn to a unit circle with centre at the origin and these tangents intersect at a point $A$ enclosing an angle of $60^o$. The area enclosed by these tangents and the arc of the circle is

If the length of the tangent from any point on the circle $(x-3)^2+(y+2)^2=5r^2$ to the circle $(x-3)^2+(y+2)^2=r^2$ is $16$ units, then the area between the two circles in sq. units is (in $\pi$)

The equation of a circle which touches the straight lines $x+y=2$ and $x-y=2$ and also touches the circle $x^2+y^2=1$ is: