All three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$? $\textbf{(A)}\ 4\qquad$ $\textbf{(B)}\ 6\qquad$ $\textbf{(C)}\ 8\qquad$ $\textbf{(D)}\ 10\qquad$ $\textbf{(E)}\ 16$
Solution
By symmetry, $B=(-q,q^2)$ and $C=(q,q^2)$ for some $q>0$.
$BC=2q$, and the height from $A=(0,0)$ to the line $y=q^2$ is $q^2$.
All three vertices of $\triangle ABC$ lie on the parabola $y=x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis. The distance from $A$ to the line $BC$ is equal to the length of $BC$. What is the area of the triangle?
By symmetry, $B=(-q,q^2)$ and $C=(q,q^2)$ for some $q>0$.