The locus of the point of intersection of the tangents to the circle $x^2+y^2=16$,such that the angle between them is $60^{\circ}$,is

The locus of the centre of the circle which cuts off intercepts of length $2a$ and $2b$ from the $x$-axis and $y$-axis respectively,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-

$A$ circle of constant radius $a$ passes through the origin $O$ and cuts the coordinate axes at points $P$ and $Q$. The equation of the locus of the foot of the perpendicular from $O$ to $PQ$ is:

The locus of the mid-points of the chords of the circle $x^2 + y^2 - 2x - 4y - 11 = 0$ which subtend $60^o$ at the centre is

The locus of the centre of the circles passing through the origin and cutting off a chord of length $2$ units on the line $x=1$ is