The locus of a point which is at a distance of $4$ units from $(3, -2)$ in the $xy$-plane is

If all chords of the curve $2x^2 - y^2 + 3x + 2y = 0$,which subtend a right angle at the origin,always pass through a fixed point $(\alpha, \beta)$,then $(\alpha, \beta) =$

Let $RS$ be the diameter of the circle $x^2+y^2=1$,where $S$ is the point $(1,0)$. Let $P$ be a variable point (other than $R$ and $S$) on the circle and tangents to the circle at $S$ and $P$ meet at the point $Q$. The normal to the circle at $P$ intersects a line drawn through $Q$ parallel to $RS$ at point $E$. Then the locus of $E$ passes through the point$(s)$:
$(A)$ $\left(\frac{1}{3}, \frac{1}{\sqrt{3}}\right)$ $(B)$ $\left(\frac{1}{4}, \frac{1}{2}\right)$ $(C)$ $\left(\frac{1}{3},-\frac{1}{\sqrt{3}}\right)$ $(D)$ $\left(\frac{1}{4},-\frac{1}{2}\right)$

Let $A=(2,0)$ and $B=(0,-2)$. Let $P$ be any point such that the sum of the distances of $P$ from $A$ and $B$ is $4$. Then the equation of the locus of the point $P$ is

Find the locus of a point such that its distance from the point $(0, -1)$ is twice its distance from the line $3x + 4y + 1 = 0$.

The locus of the centre of a circle passing through $(a, b)$ and cutting orthogonally to the circle $x^2 + y^2 = p^2$ is