$A$ point $P$ moves on the line $2x - 3y + 4 = 0$. If $Q(1, 4)$ and $R(3, -2)$ are fixed points,then the locus of the centroid of $\Delta PQR$ is a line

$A$ frog is presently located at the origin $(0,0)$ in the $XY$-plane. It always jumps from a point with integer coordinates to a point with integer coordinates,moving a distance of $5$ units in each jump. What is the minimum number of jumps required for the frog to go from $(0,0)$ to $(0,1)$?

If non-zero numbers $a, b, c$ are in harmonic progression,then the straight line $\frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0$ always passes through a fixed point. Find the coordinates of that point.

The Cartesian form of the polar equation $\theta = \tan^{-1} 2$ is

Suppose $P$ and $Q$ are the midpoints of the sides $AB$ and $BC$ of a triangle where $A(1, 3)$,$B(3, 7)$,and $C(7, 15)$ are vertices. Then the locus of $R$ satisfying $AC^2 + QR^2 = PR^2$ is

$A(1, 0)$,$B(0, 2)$,and $C(1, 2)$ are three points on the $XY$-plane. If a point $P(x, y)$ moves such that the area of $\triangle PAB$ is twice the area of $\triangle ABC$,then the locus of the point $P$ is: