The length of the shortest path that begins at the point $(2, 5)$,touches the $x-$axis,and then ends at a point on the circle $x^2 + y^2 + 12x - 20y + 120 = 0$.

Two circles with equal radii intersect at the points $(0, 1)$ and $(0, -1).$ The tangent at the point $(0, 1)$ to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is

In the figure,$ABCD$ is a unit square. $A$ circle is drawn with centre $O$ on the extended line $CD$ and passing through $A$. If the diagonal $AC$ is tangent to the circle,then the area of the shaded region is

What is the value of the mathematical constant $\pi$?

Which of the following statements are true and which are false? In each case,give a valid reason for your answer.
$p:$ Each radius of a circle is a chord of the circle.

Three circles of radii $a, b, c$ $(a < b < c)$ touch each other externally. If they have the $x$-axis as a common tangent,then: