Suppose that two people, $A$ and $B$, walk along the parabola $y=x^{2}$ in such a way that the line segment $L$ between them is always perpendicular to the line tangent to the parabola at $A$ 's position. What are the positions of $A$ and $B$ when L has minimum length?
a. Assume that $A$ 's position is $\left(a, a^{2}\right),$ where $a>0 .$ Find the slope of the line tangent to the parabola at $A$ and find the slope of the line that is perpendicular to the tangent line at $A.$
b. Find the equation of the line joining $A$ and $B$ when $A$ is at $\left(a, a^{2}\right).$
c. Find the position of $B$ on the parabola when $A$ is at $\left(a, a^{2}\right).$
d. Write the function $F(a)$ that gives the square of the distance between $A$ and $B$ as it varies with $a$. (The square of the distance is minimized at the same point that the distance is minimized; it is easier to work with the square of the distance.)
e. Find the critical point of $F$ on the interval $a > 0.$
f. Evaluate $F$ at the critical point and verify that it corresponds to an absolute minimum. What are the positions of $A$ and $B$ that minimize the length of $L ?$ What is the minimum length?
g. Graph the function $F$ to check your work.