Question

The letters r and Θ represent polar coordinates. perform the following.

r = 5/r-4cosΘ

cross multiply then convert the polar equation to rectangular form

Answer 1

`\boxed{(x - (1 )/( 2))^(2) + y^(2) = (21 )/( 4)}\\`

Explanation:

`r = (5)/(r - cos\theta)\\r^(2) - rcos\theta = 5\\\\`

`\text{Use the relationships}\\r^(2) = x^(2) + y^(2); cos\theta = ( x)/(r )\\`

`\begin{array}{ll}x^(2) + y^(2) - r((x )/(r )) = 5 & \text{Made the substitutions } \\x^(2) + y^(2) - x = 5 & \text{Simplified } \\x^(2) -x + y^(2) = 5 & \text{Rearranged } \\x^(2) -x + (1 )/( 4) + y^(2)= 5 + (1 )/( 4) & \text{Completed the square } \\(x - (1 )/( 2))^(2) + y^(2)= (21 )/( 4) & \text{Wrote as sum of squares} \\\end{array}\\\\`

`\text{This is the equation of a circle with centre at (0.5, 0 ) and radius equal to}\\(√(21) )/( 2) \approx 2.291\\`

`\boxed{(x - (1 )/( 2))^(2) + y^(2)= (21 )/( 4)}\\`