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 $...... \, km$.

The circle $x^2 + y^2 - 4x - 4y + 4 = 0$ is...

Choose the incorrect statement about the two circles whose equations are given below:
$x^{2}+y^{2}-10x-10y+41=0$ and $x^{2}+y^{2}-16x-10y+80=0$

Let $y=x$ be the equation of a chord of the circle $C_{1}$ (in the closed half-plane $x \ge 0$) of diameter $10$ passing through the origin. Let $C_{2}$ be another circle described on the given chord as its diameter. If the equation of the chord of the circle $C_{2}$, which passes through the point $(2, 3)$ and is farthest from the center of $C_{2}$, is $x+ay+b=0$, then $a-b$ is equal to:

If $P$ is a point on the circle $x^{2}+y^{2}=4$, $Q$ is a point on the straight line $5x+y+2=0$ and $x-y+1=0$ is the perpendicular bisector of $PQ$, then $13$ times the sum of the abscissae of all such points $P$ is ........... .

The equation of the circle with $(1,1)$ as centre and which cuts a chord of length $4 \sqrt{2}$ units on the line $x+y+1=0$ is