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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

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

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

A circle of radius $13$ cm touches the adjacent sides AB and BC of a square ABCD at M and N, respectively. If AB = $18$ cm and the circle intersects the other two sides CD and DA at P and Q, respectively, then the area, in sq. cm, of triangle PMD is

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

If the shortest distance of a given point to a given circle is $4 \, \text{cm}$ and the longest distance is $9 \, \text{cm}$, then the radius of the circle is

If $ \theta $ is the angle between the pair of tangents drawn from the point $(0,\frac{7}{2})$ to the circle $x^2 + y^2 - 14x + 16y + 88 = 0$, then $\tan \theta$ equals

Which of the following straight lines are both tangent to the circle $x ^ 2 + y ^ 2 - 6x + 4y - 12 = 0$?

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

The shortest distance from the point $(-4,3)$ to the circle $(x^2 + y^2 = 1)$ is __________.

A chord is drawn inside a circle, such that the length of the chord is equal to the radius of the circle. Now, two circles are drawn, one on each side of the chord, each touching the chord at its midpoint and the original circle. Let k be the ratio of the areas of the bigger inscribed circle and the smaller inscribed circle, then k equals

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

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