- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
14.21 Miscellaneous Review Exercises
- Prove that an equation of the line that is tangent to the parabola $y^2 = 4cx$ at the point $(x_1, y_1)$ can be written in the form $y_1 y = 2c(x + x_1).$
- Express slope $m$ and $y$-intercept $b$ in terms of the coordinates of the point of contact $(x_1, y_1).$
- Recall from Exercise 3 that the line of slope $m$ is tangent to the parabola with equation $y^2 = 4cx$ at the point \begin{align*} \\ \left(\frac{c}{m^2}, \frac{2c}{m}\right) \end{align*} With $y$-intercept $b = c/m.$ If we denote the point of contact as $(x_1, y_1),$ then we can rewrite $m$ in terms of $y_1,$ and $b$ in terms of $x_1.$ For $y_1 \neq 0,$ the equation of the line then becomes \begin{align*} \\ y &= mx + b \\ \\ &= \frac{2c}{y_1}x + mx_1 \\ \\ &= \frac{2c}{y_1}\left(x + x_1\right) \end{align*} Multiplying both sides by $y_1$ gives us: \begin{align*} y_1 y &= 2c(x + x_1) \quad \blacksquare \end{align*}