- 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
- Two tangent lines are drawn to the ellipse $x^2 + 4y^2 = 8,$ each parallel to the line $x + 2y = 7.$ Find the points of tangency.
- Use the result of Exercise 11.
- In Exercise 11, we showed that a line tangent to the ellipse $x^2/a^2 + y^2/b^2 = 1$ at $(x_0, y_0)$ has the equation \begin{align*} \\ \frac{x}{a^2}x_0 + \frac{y}{b^2}y_0 &= 1 \end{align*} From the given ellipse equation, we know that $a^2 = 8$ and $b^2 = 2.$ As such, we wish to find $(x_0, y_0)$ such that $(x_0/8)\mathbf{i} + (y_0/2)\mathbf{j}$ is parallel to $\mathbf{i} + 2\mathbf{j}$ and $x_0^2 + 4y_0^2 = 8.$ In other words, we wish to find $x_0$ and $y_0$ such that $x_0 = 2y_0$ and that $4y_0^2 = 8 - x_0^2.$ Substituting $x_0^2 = 4y_0^2$ we see that $y_0$ is satisfied by $1$ or $-1$ Then, plugging in $x_0 = 2y_0$ we can see that the two points of tangency are: \begin{align*} (x_0, y_0) &= (2, 1), \quad (x_0, y_0) = (-2, -1) \quad \blacksquare \end{align*}