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

Let $AB$ be the diameter of a semicircle $S$. The locus of the centres of circles which are tangent to $AB$ and to $S$ is an arc of

The locus of the centre of the circle touching the $x$-axis and passing through the point $(-1, 1)$ is

The locus of the centre of a circle which passes through the point $(a, 0)$ and touches the line $x + 1 = 0$ is

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:

Let $C$ be the circle with centre $(0,0)$ and radius $3$ units. The equation of the locus of the midpoints of the chords of the circle $C$ that subtend an angle of $\frac{2\pi}{3}$ at its centre is: