Question

Convert each polar equation into a rectangular equation.

$r=\frac{10}{5+4\cos \left(\theta \right)}$

Short answer

The rectangular equation is

$9x^2+25y^2+80x=100$

Step-by-step solution

Step 1. Given information

The polar equation is

$r=\frac{10}{5+4\cos \left(\theta \right)}$

Step 2. Convert into the polar equation

The polar equation is

$r=\frac{10}{5+4\cos \left(\theta \right)}$

Multiply both sides by the denominator

$$\begin{gathered} r\times (5+4\cos (\theta \left)\right)=\frac{10}{5+4\cos \left(\theta \right)}\times 5+4\cos \left(\theta \right) \\ r\times (5+4\cos (\theta \left)\right)=10 \\ 5r+4r\cos \left(\theta \right)=10 \end{gathered}$$

Apply conversion equation formula

$r=\sqrt{x^2+y^2},x=r\cos \left(\theta \right)$

$5\sqrt{x^2+y^2}+4x=10$

Subtract both sides by $4x$

$$\begin{gathered} 5\sqrt{x^2+y^2}+4x-4x=10-4x \\ 5\sqrt{x^2+y^2}=10-4x \end{gathered}$$

Take square on both sides

$$\begin{gathered} {\left(5\sqrt{x^2+y^2}\right)}^2={(10-4x)}^{2}25(x^2+y^2)=100-80x+16x^2 \\ 25x^2+25y^2=100-80x+16x^2 \\ 25x^2+25y^2+80x-16x^2=100 \\ 9x^2+25y^2+80x=100 \end{gathered}$$

So, the rectangular equation is

$9x^2+25y^2+80x=100$