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.
- (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).
- (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.
- (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. - (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.
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