Question

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

Answer 1

Answer:

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

Step-by-step explanation:

$r = \frac{5}{r - cos\theta}\\r^{2} - rcos\theta = 5$

$\text{Use the relationships}\\r^{2} = x^{2} + y^{2}; cos\theta = \frac{ x}{r }$

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

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

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