The locus of a point which moves in such a way that its distance from $(0,0)$ is three times its distance from the $x$-axis is given by:

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

The equation of the locus of the midpoints of the chords of the circle $4x^2 + 4y^2 - 12x + 4y + 1 = 0$ that subtend an angle of $\frac{2\pi}{3}$ at its center is

The locus of the centre of a circle which passes through two fixed points $(a, 0)$ and $(-a, 0)$ is

$A$ variable circle is drawn passing through the origin $O$. It intersects the $X$ and $Y$ axes at points $A$ and $B$ respectively,such that $OA + 2OB = K$ (a non-zero constant). The circle always passes through a fixed point $P$ other than the origin. The point $P$ lies on -

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-