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 of that quadrilateral is $A.$ What is $\frac{A}{4}?$

If $ABCD$ is a $2\times 2$ square, $E$ is the midpoint of $\overline{AB}, F$ is the midpoint of $\overline{BC}, \overline{AF}$ and $\overline{DE}$ intersect at $I$ and $BD$ and $AF$ intersect at $H$ and the area of the quadrilateral $BEIH$ is $\frac{k}{2k+1}$, find $k.$

Locus of the points which are at equal distances from $3 x+4 y-11=0$ and $12 x+5 y+2=0$ and which is near to the origin is $ax+by-133=0$ then $a+b$ is equal to

The distance between the pair of parallel lines $x^2+2xy+y^2-8ax-8ay-9a^2=0$ is $\left(ka\right)$ units then $k^2$ is equal to -