MATH SOLVE

3 months ago

Q:
# the letters r and Θ represent polar coordinates. perform the following.r = 5/r-4cosΘcross multiply then convert the polar equation to rectangular form

Accepted Solution

A:

Answer:[tex]\boxed{(x - \frac{1 }{ 2})^{2} + y^{2} = \frac{21 }{ 4}}\\[/tex]Step-by-step explanation:[tex]r = \frac{5}{r - cos\theta}\\r^{2} - rcos\theta = 5\\\\[/tex][tex]\text{Use the relationships}\\r^{2} = x^{2} + y^{2}; cos\theta = \frac{ x}{r }\\[/tex][tex]\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}\\\\[/tex][tex]\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\\[/tex][tex]\boxed{(x - \frac{1 }{ 2})^{2} + y^{2}= \frac{21 }{ 4}}\\[/tex]