Chapter 9.2, Problem 62AYU

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.

$r=8\sin \theta ;r=4\csc \theta$

To determine

To find: Graph the polar equations $r = 8 \text{sin} \theta ; r = 4 \csc \theta$ on the same polar grid and find the point of intersections if any.

Answer to Problem 62AYU

Solution

The graph of the polar equations $r = 8 \text{sin} \theta and r = 4 \csc \theta$ are equation of circle and straight horizontal line at 4.

From the above graph, the point of intersections are $\left(- 4, 4\right) \text{and} \left(4, 4\right)$.

Given:

It is asked to graph the polar equations $r = 8 \text{sin} \theta ; r = 4 \csc \theta$.

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 polar axis, center on the line $\theta = \frac{\pi }{2}$, radius $a$ is given by the equation

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

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

This polar equation can be written as

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

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

Consider $r = 4 \csc \theta$,

$r = 4 \csc \theta = 4\frac{1}{\text{sin} \theta }$

$r \text{sin} \theta = 4$

Which is nothing but the horizontal line at $4$.

To sketch:

$r = 8 \text{sin} \theta ; r = 4 \csc \theta$.

From the above graph, the point of intersections are $\left(-4, 4\right) \text{and} \left(4, 4\right)$.