The locus of the midpoints of the chords of the circle $x^2-2x+y^2=0$ drawn from the point $(0,0)$ on it is

Let $x$ and $y$ be real numbers satisfying the equation $x^2 - 4x + y^2 + 3 = 0$. If the maximum and minimum values of $x^2 + y^2$ are $M$ and $m$ respectively,then the numerical value of $M - m$ is:

If $P$ is a variable point such that the sum of the distances from $P$ to the points $A(2,2)$ and $B(2,-2)$ is $4$,then the locus of $P$ represents

Let the locus of the centre $(\alpha, \beta)$,$\beta > 0$,of the circle which touches the circle $x^{2} + (y - 1)^{2} = 1$ externally and also touches the $x$-axis be $L$. Then the area bounded by $L$ and the line $y = 4$ is.

$A$ circle touches the $x$-axis and cuts off a chord of length $2l$ from the $y$-axis. The locus of the centre of the circle is

Let $P$ be a variable point on a circle $C$ and $Q$ be a fixed point outside $C$. If $R$ is the midpoint of the line segment $PQ$, then the locus of $R$ is