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

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

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

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

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

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

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