If the points $A(2,3)$ and $B(3,2)$ form a triangle with a variable point $P(t, t^2)$,where $t$ is a parameter,then the equation of the locus of the centroid of triangle $ABP$ is:

If the angle between a pair of tangents drawn from a point $P$ to the circle $x^2+y^2+4x-6y+9 \sin^2 \alpha+13 \cos^2 \alpha=0$ is $2 \alpha$,then the equation of the locus of $P$ is

Let the locus of the centre $(\alpha, \beta)$,$\beta > 0$,of the circle which touches the circle $x^{2} + (y - 1)^{2} = 1$ externally and also touches the $x$-axis be $L$. Then the area bounded by $L$ and the line $y = 4$ is.

If $A(2, 3)$ and $B(2, -3)$ are two points,then the equation of the locus of a point $P$ such that $PA + PB = 8$ is

$A(a, 0)$ is a fixed point and $\theta$ is a parameter such that $0 < \theta < 2 \pi$. If $P(a \cos \theta, a \sin \theta)$ is a point on the circle $x^2+y^2=a^2$ and $Q(b \sin \theta, -b \cos \theta)$ is a point on the circle $x^2+y^2=b^2$,then the locus of the centroid of the triangle $APQ$ is

From the point $P(3, 4)$,chords are drawn to the circle $x^2 + y^2 - 4x = 0$. Find the locus of the midpoints of these chords.