If $P_1, P_2, P_3, \ldots, P_n$ are $n$ points on the line $y=x$ all lying in the first quadrant,such that $OP_n = n(OP_{n-1})$ ($O$ is origin),$OP_1 = 1$ and $P_n = (2520 \sqrt{2}, 2520 \sqrt{2})$,then $n=$

The locus of a point which moves such that the area of the triangle formed by it with the vertices $(1, 2)$ and $(-2, 5)$ is $8$ sq. units is/are

$A$ moving line intersects the lines $x+y=0$ and $x-y=0$ at the points $A$ and $B$ respectively, such that the area of the triangle with vertices $(0,0)$, $A$, and $B$ has a constant area $C$. The locus of the mid-point of $AB$ is given by the equation:

Every point $(x, y)$ on the curve $3x + 2y - 3xy = 0$ is the centroid of a triangle formed by the coordinate axes and a line $(L)$ intersecting both the coordinate axes. Then all such lines $(L)$

If the point $\left(\alpha, \frac{7 \sqrt{3}}{3}\right)$ lies on the curve traced by the mid-points of the line segments of the lines $x \cos \theta + y \sin \theta = 7, \theta \in \left(0, \frac{\pi}{2}\right)$ between the coordinate axes,then $\alpha$ is equal to

Let $A(2, 3)$ and $B(-4, 5)$ be two fixed points. If a point $P(x, y)$ moves such that the area of $\Delta PAB = 12$ square units,find the locus of $P$.