Let ABCD be a rectangle with AB = 72 cm and BC = 30 cm. A circle passing through points A and C cuts the side AB at P such that AP = 56 cm. The radius, in cm, of the circle is

The circle $x^2 + y^2 - 6x - 10y + k = 0$ does not touch or intersect the coordinate axes. If the point $(1, 4)$ does not lie outside the circle, and the range of $k$ is $(a, b]$, then $a + b$ is

A circle touches the y-axis at $(0, 4)$ and passes through the point $(-2, 0)$. Then the radius of the circle is

In the xy-plane let $A = (- 2, 0), B = (2, 0)$ . Define the set S as the collection of all points C on the circle $x ^ 2 + y ^ 2 = 4$ such that the area of the triangle ABC is an integer. The number of points in the set S is

A person walks one lap along a circle at a speed $v$. Thereafter, he runs one lap along the boundary of the largest square that can be inscribed in the circle at a speed $3v$. The ratio of the time he walks to the time he runs is

On a circular path of radius 6 m a boy starts from a point $A$ on the circumference and walks along a chord $AB$ of length 3 m. He then walks along another chord $BC$ of length 2 m to reach point $C$. The point $B$ lies on the minor arc $AC$. The distance between point $C$ from point $A$ is

Let the circle $x^2 + y^2 = 2ax + 2by$ intersect the x-axis at point $A(\alpha, 0)$ and y-axis at point $B(0, \beta)$, where $\alpha\beta \neq 0$. If the point $C(p, q)$ lies on the chord AB, then $\frac{p+\alpha}{a} + \frac{q+\beta}{b}$ equals