Chapter 10: Q 5. (page 683)

True or False:If $(r, θ)$ are polar coordinates, the equation $r = \frac{2}{2 + 3 \sin (θ)}$ defines a hyperbola.

Short Answer

Expert verified

This statement is True.

Step by step solution

Step 1. Given information

The polar equation is

$r = \frac{2}{2 + 3 \sin (θ)}$

Step 2. Checking the statement

The polar equation is

$r = \frac{2}{2 + 3 \sin (θ)}$

Divide top and bottom by $2$

$r = \frac{\frac{2}{2}}{\frac{2}{2} + \frac{3}{2} \sin (θ)} r = \frac{1}{1 + \frac{3}{2} \sin (θ)}$

Compare with the standard polar equation

$r = \frac{e p}{1 + e \sin (θ)}$

So,

$e = \frac{3}{2} > 1$

Therefore, this polar equation is a hyperbola.

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True or False:If $(r, θ)$ are polar coordinates, the equation $r = \frac{2}{2 + 3 \sin (θ)}$ defines a hyperbola.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Checking the statement

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True or False:If $(r, θ)$ are polar coordinates, the equation $r = \frac{2}{2 + 3 \sin (θ)}$ defines a hyperbola.