$A_{1}, A_{2},....A_{n}$ are points on the line $y=x$ lying in the positive quadrant such that $OA_{n}=n O \ A_{n-1}, O$ being the origin . If $OA_{1}=1$ then the coordinates of $A_{8}$ are $(3a \sqrt 2, 3a \sqrt 2)$ where $a$ is equal to

If $a$ and $b$ are real numbers between $0$ and $1$ such that the points $(a,1),(1,b)$ and $(0,0)$ from an equilateral triangle then the values of $a$ and $b$ respectively ?

Let the length of the sides of a triangle $\triangle ABC$ be integers with $A$ as the origin. $(2, -1)$ and $(3, 6)$ are points on the line $AB$ and $AC$ respectively (line $AB$ and $AC$ may be extended to contain these points), and lengths of exactly two sides are primes that differ by $50$. If $a$ is least possible length of the third side and $S$ is the least possible perimeter of the triangle, then $aS$ is equal to

An equilateral triangle has one vertex at $(3, 4)$ and another at $(-2, 3)$. Find the coordinates of the third vertex.