Chapter 9.2, Problem 61AYU

In Problems 63-68, graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph.

$r=8\cos \theta ;r=2\sec \theta$

To determine

To find: Graph the polar equations $r = 8 \text{cos} \theta ; r = 2 \sec \theta$ on the same polar grid and find the point of intersections if any.

Answer to Problem 61AYU

Solution

The graph of the polar equations $r = 8 \text{cos} \theta and r = 2 \sec \theta$ are equation of circle and straight vertical line parallel to $y$-axis.

From the above graph, the point of intersections are (2, 3.464) and (2, -3.464).

Given:

It is asked to graph the polar equations $r = 8 \text{cos} \theta ; r = 2 \sec \theta$.

Explanation of Solution

Formula to be used:

$\left(r \text{cos} \theta ,r \text{sin} \theta \right)=\left(x, y\right)$

Equation of circle passing through the pole, tangent to the line $\theta = \frac{\pi }{2}$, center on the polar axis, radius $a$ is given by the equation

$r = \pm 2a \text{cos} \theta$, $a > 0$

Consider $r = 8 \text{cos} \theta$,

This polar equation can be written as

$r = 2\left(4\right) \text{cos} \theta$

The above equation is an equation of circle passing through the pole, tangent to the line $\theta = \frac{\pi }{2}$, center on the polar axis, radius $4$.

Consider $r = 2 \sec \theta$,

$r = 2 \sec \theta =2 \frac{1}{\text{cos} \theta }$

$r \text{cos} \theta = 2$

Which is nothing but the vertical straight line at 2.

To sketch:

$r = 8 \text{cos} \theta ; r = 2 \sec \theta$

From the above graph, the point of intersections are (2, 3.464) and (2, -3.464).