The locus of a point $P$ which divides the line joining $(1, 0)$ and $(2\cos \theta, 2\sin \theta)$ internally in the ratio $2 : 3$ for all $\theta$ is a

The locus of the midpoint of the chord of contact of tangents drawn from a point on the line $4x - 5y = 20$ to the circle $x^2 + y^2 = 9$ is:

The locus of the midpoints of the chords of the circle $x^2+y^2=16$,which are tangents to the hyperbola $9x^2-16y^2=144$,is

The set of all points that are at a distance of at least $2$ units from $(-3, 0)$ 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-

The equation of the locus of the centroid of the triangle whose vertices are $(a \cos k, a \sin k)$,$(b \sin k, -b \cos k)$ and $(1, 0)$,where $k$ is a parameter,is