- 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
- Show that the vectors $T = (y/b^2)\,\mathbf{i} + (x/a^2)\,\mathbf{j}$ and $N = (x/a^2)\,\mathbf{i} - (y/b^2)\,\mathbf{j}$ are, respectively, tangent and normal to the hyperbola $x^2/a^2 - y^2/b^2 = 1$ if placed at the point $(x, y)$ on the curve.
- Use the same approach as in Exercise 10.
- If we parameterize the point $(x, y)$ by $x,$ then its derivative vector is $D = \mathbf{i} + \frac{dy}{dx}\mathbf{j}.$ To show that $T$ is tangent to the curve at $(x, y),$ then we must show that it is parallel to $D$ at $(x, y).$ The derivative of the hyperbola at $(x, y)$ satisfies the following relation \begin{align*} \frac{x}{a^2} &= \frac{y}{b^2}\frac{dy}{dx} \end{align*} But, setting $c = y/b^2,$ we can see that $T = cD.$ This shows that $T$ is parallel to the line tangent to $(x, y).$ To make $T$ tangent to the curve, we can simlpy translate it by $(x, y).$ To show that $N$ is normal to the hyperbola at $(x, y),$ it will suffice to show that $N \cdot T = 0$ and then to translate it by $(x, y). \quad \blacksquare$