I was solving some problems on parabola. I saw a question and solved it, but my solution was way too big. The question was:
If $$\left(\frac{a}{b}\right)^{1/3}+\left(\frac{b}{a}\right)^{1/3} = \frac{\sqrt{3}}{2}$$ Then angle of intersection of parabola $y^2=4ax$ and $x^2=4by$ at a point other than origin is $\ldots$
My solution was too lengthy. I found the point of intersection of parabolas , got tangents, and then found angle as $\dfrac{\pi}{3}$ . Can anybody suggest a shorter method ?
Use the optical properties of the parabola. Let $\Gamma$ be a parabola with vertex in $V$ and let $P\neq V$ a point on $\Gamma$, $l$ the tangent to $\Gamma$ in $P$. Given that $P_\perp$ is the projection of $P$ on the axis $a$ of the parabola, we have that $a\cap l$ is just the symmetric of $P_\perp$ with respect to $V$.
With a few calculations, this just gives that your first formula is the sine of the angle between the tangents.