If the locus of the point,whose distances from the point $(2,1)$ and $(1,3)$ are in the ratio $5:4$,is $ax^2+by^2+cxy+dx+ey+170=0$,then the value of $a^2+2b+3c+4d+e$ is equal to:

If ${\theta _1}$ and ${\theta _2}$ are the inclinations of the tangents drawn from a point $P(h, k)$ to the circle ${x^2} + {y^2} = {a^2}$ with the $x$-axis,then the locus of $P$,given that $\cot {\theta _1} + \cot {\theta _2} = c$,is:

$A$ line segment $AM = a$ moves in the $XOY$ plane such that $AM$ is parallel to the $X$-axis. If $A$ moves along the circle $x^2 + y^2 = a^2$,then the locus of $M$ is

If the locus of the mid-point of the line segment from the point $(3, 2)$ to a point on the circle $x^{2} + y^{2} = 1$ is a circle of radius $r$,then $r$ is equal to:

Let $A = (0, 4)$ and $B = (2 \cos \theta, 2 \sin \theta)$,for some $0 < \theta < \frac{\pi}{2}$. Let $P$ divide the line segment $AB$ in the ratio $2:3$ internally. The locus of $P$ is

The locus of the centre of a circle,which touches two given circles externally,is