Products
Rewards
from HOLOOLY

We are determined to provide the latest solutions related to all subjects FREE of charge!

Enjoy Limited offers, deals & Discounts by signing up to Holooly Rewards Program

HOLOOLY

HOLOOLY
TABLES

All the data tables that you may search for.

HOLOOLY
ARABIA

For Arabic Users, find a teacher/tutor in your City or country in the Middle East.

HOLOOLY
TEXTBOOKS

Find the Source, Textbook, Solution Manual that you are looking for in 1 click.

HOLOOLY
HELP DESK

Need Help? We got you covered.

## 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.

## 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).