Basics of 2D Coordinate Geometry

From the point $A\left(0,3\right)$ on the circle $x^2+4x+{\left(y-3\right)}^2=0$, a chord $AB$ is drawn and extended to a point $M$ such that $AM=2AB$. The equation of the locus of $M$ is

Square $ABCD$ has side length $30.$ Point $P$ lies inside the square so that $AP=12$ and $BP=26.$ The centroids of $∆ABP, ∆BCP, \Delta CDP$ and $∆DAP$ are the vertices of a convex quadrilateral. The area (in sq. units) of the quadrilateral is

Point $P$ divides line $AB$, with $A\left(5,14\right)$ and $B\left(7,16\right)$ in ratio $3:2$ externally. Find the coordinates of $P$.

Point $P$ divides line $AB$, with $A\left(2,3\right)$ and $B\left(5,6\right)$ in ratio $3:2$ externally. Find the coordinates of $P$.

Point $P$ divides line $AB$, with $A\left(7,6\right)$ and $B\left(5,10\right)$ in ratio $3:2$ externally. Find the coordinates of $P$.

Point $P$ divides line $AB$, with $A\left(5,4\right)$ and $B\left(8,6\right)$ in ratio $3:2$ externally. Find the coordinates of $P$.

Point $P$ divides line $AB$, with $A\left(3,4\right)$ and $B\left(7,12\right)$ in ratio $3:2$ externally. Find the coordinates of $P$.

Write the quadrant to which the following point belongs to.

$\left(-5,2\right)$

Write the quadrant to which the following point belongs to.

$\left(-7,4\right)$

Write the quadrant to which the following point belongs to.

$\left(-5,-3\right)$

Find the foot of perpendicular from $\left(5,8\right)$ to $x-$axis.

Find the foot of perpendicular from $\left(8,6\right)$ to $y-$axis.

Find the foot of perpendicular from $\left(9,7\right)$ to $x-$axis.

Find the foot of perpendicular from $\left(4,5\right)$ to $y-$axis.

Find the foot of perpendicular from $\left(2,3\right)$ to $x-$axis.

Define $2\mathrm{D}$ coordinate system. Find the quadrant $(4,-2)$ belongs to.

Define $2\mathrm{D}$ coordinate system. Find the quadrant $(1,-2)$ belongs to.

Define $2\mathrm{D}$ coordinate system. Find the quadrant $(-1,-2)$ belongs to.

Define $2\mathrm{D}$ coordinate system. Find the quadrant $(-1,2)$ belongs to.

If the equation of a line which divides the line segment joining the points $\left(1,0\right)$ and $\left(3,0\right)$ in the ratio $2:1$ and is also perpendicular to it, is $ax=7$, then the value of $a$ is equal to