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Chapter 2

Q. 2.4

Find all points at which the curve x=2+7 \cos \theta, y=8+3 \sin \theta has a vertical or horizontal tangent.

Step-by-Step

Verified Solution

We find that f_{1}^{\prime}(\theta)=x^{\prime}(\theta)=-7 \sin \theta, which is 0 when \theta=n \pi for some integer n. If \theta=n \pi, then

x=\left\{\begin{array}{ll}9, & n \text { even } \\-5, & n \text { odd }\end{array} \quad \text { and } \quad y=8\right.

so there are vertical tangents at (9, 8) and (-5,8). Similarly, f_{2}^{\prime}(\theta)=y^{\prime}(\theta)=3 \cos \theta, which is 0 when \theta=\left(n+\frac{1}{2}\right) \pi for some integer n. If \theta=\left(n+\frac{1}{2}\right) \pi, then

x=2 \quad \text { and } \quad y= \begin{cases}11, & n \text { even } \\ 5, & n \text { odd }\end{cases}

so the curve has horizontal tangents at (2, 11) and (2, 5). You should verify that when f_{1}^{\prime}(\theta)=0, f_{2}^{\prime}(\theta) \neq 0, and vice versa.

REMARK.    It is not difficult to verify that x and y satisfy the equation

\frac{(x-2)^{2}}{49}+\frac{(y-8)^{2}}{9}=1.

This is the equation of an ellipse centered at (2, 8).