$A$ variable chord is drawn through the origin to the circle $x^2 + y^2 - 2ax = 0$. The locus of the centre of the circle drawn on this chord as diameter is:

Let the function $f(x) = 2x^{2} - \log_{e} x$,$x > 0$,be decreasing in $(0, a)$ and increasing in $(a, 4)$. $A$ tangent to the parabola $y^{2} = 4ax$ at a point $P$ on it passes through the point $(8a, 8a - 1)$ but does not pass through the point $(-\frac{1}{a}, 0)$. If the equation of the normal at $P$ is $\frac{x}{\alpha} + \frac{y}{\beta} = 1$,then $\alpha + \beta$ is equal to-

What is the equation of the locus of a point which moves such that $4$ times its distance from the $x$-axis is the square of its distance from the origin?

Let $A = (a, 0)$ and $B = (-a, 0)$ be two fixed points. For $a \in (-\infty, 0)$,point $P(x, y)$ moves in the plane such that $PA = nPB$ $(n \neq 0, n \neq 1)$. If $0 < n < 1$,then which of the following is true regarding the locus of $P$?

If $A=(1,2)$,$B=(2,1)$ and $P$ is any point satisfying the condition $PA+PB=3$,then the equation of the locus of $P$ 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 $(\alpha, \beta) =$