The locus of the centers of the circles,which have the same area and have $3x - 4y + 4 = 0$ and $6x - 8y - 7 = 0$ as their common tangents,is

The locus of the centre of circles which pass through $(0, 1)$ and touch the line $y = x$ is -

If $A(\cos \alpha, \sin \alpha)$,$B(\sin \alpha, -\cos \alpha)$,and $C(1, 2)$ are the vertices of a $\triangle ABC$,then the locus of its centroid is:

$A$ point moves such that its distance from the point $(4, 0)$ is half that of its distance from the line $x = 16$. The locus of this point is

$A$ variable chord is drawn through the origin to the circle $x^2 + y^2 - 2ax = 0$. The locus of the centre of the circle drawn on this chord as diameter is:

Let $A$ be the point $(1, 2)$ and $B$ be any point on the curve $x^2 + y^2 = 16$. If the centre of the locus of the point $P$,which divides the line segment $AB$ in the ratio $3:2$ is the point $C(\alpha, \beta)$,then the length of the line segment $AC$ is