Chapter 11, Problem 5P

Tangent Lines to a Parabola In this problem we show that the line tangent to the parabola y = x² at the point (a, a²) has the equation y = 2ax − a².

1. (a) Let m be the slope of the tangent line at (a, a²). Show that the equation of the tangent line is y − a² = m(x − a).
2. (b) Use the fact that the tangent line intersects the parabola at only one point to show that (a, a²) is the only solution of the system.

$\{\begin{array}{l}y-a^2=m(x-a)\hfill \\ y=x^2\hfill \end{array}$

1. (c) Eliminate y from the system in part (b) to get a quadratic equation in x. Show that the discriminant of this quadratic is (m − 2a)². Since the system in part (b) has exactly one solution, the discriminant must equal 0. Find m.
2. (d) Substitute the value for m you found in part (c) into the equation in part (a), and simplify to get the equation of the tangent line.