Prove that the circle described on a focal lengOf a parabola as diameter touches the tangent and vertes
First, what do you mean by "described on a focal length of a parabola"? Do you mean having the focus as center and passing through the vertex of a parabola? And what do you mean by "touches the tangent and vertex"? If the circle has the focus as center and focal length as radius then it passes through the vertex but what is the "tangent" you refer to?
The parabola $\displaystyle y= 4px^2$ has vertex at (0, 0) and focus at (0, p). The circle with center (0, p) and radius p has equation $\displaystyle x^2+ (y- p)^2= p^2$. Those two graphs intersect when $\displaystyle y= 4px^2= p- \sqrt{p^2- x^2}$. Write that as $\displaystyle \sqrt{p^2- x^2}= p- 4px^2$ and square both sides: $\displaystyle p^2- x^2= p^2- 8p^2x^2+ 16p^2x^4$ or $\displaystyle 16p^2x^4- (8p^2- 1)x^2= x^2(16p^2x^2- (8p^2- 1))= 0$. That has x= 0 as a double root and $\displaystyle x= \pm\sqrt{\frac{8p^2- 1}{16p^2}}$ are the other two roots.