Suppose that the earth is a sphere of radius $6400 \, km$. The height from the earth's surface from where exactly a fourth of the earth's surface is visible,is $...... \, km$.

If a circle of unit radius is divided into two parts by an arc of another circle subtending an angle $60^o$ on the circumference of the first circle,then the radius of the arc is

The line $2x - 3y = 1$ divides the circular region $x^2 + y^2 \leq 6$ into two parts. If $S = \left\{ \left(2, \frac{3}{4}\right), \left(\frac{5}{2}, \frac{3}{4}\right), \left(\frac{1}{4}, -\frac{1}{4}\right), \left(\frac{1}{8}, \frac{1}{4}\right) \right\}$,then the number of points in the set $S$ that lie inside the smaller part is:

Let $x_0, y_0$ be fixed real numbers such that $x_0^2+y_0^2 > 1$. If $x, y$ are arbitrary real numbers such that $x^2+y^2 \leq 1$,then the minimum value of $(x-x_0)^2+(y-y_0)^2$ is

Let a circle $C: (x-h)^{2} + (y-k)^{2} = r^{2}, k > 0$,touch the $x$-axis at $(1, 0)$. If the line $x + y = 0$ intersects the circle $C$ at $P$ and $Q$ such that the length of the chord $PQ$ is $2$,then the value of $h + k + r$ is equal to

If the circle $x^2 + y^2 - 6x - 8y + (25 - a^2) = 0$ touches the $x$-axis,then $a$ equals