Find the center and radius of the circle whose equation in polar coordinates is r =12sin$\theta$.
A) radius r = 6; center (-6, 6)
B) radius r = 6; center (0, -6)
C) radius r = 6; center (0, 6)
D) radius r = 6; center (6, -6)

Question

Quizplus · Accepted Answer

The given polar equation is $r = 12\sin(\theta)$. To find the center and radius in Cartesian coordinates, we convert this to $(x, y)$ form. The conversion formulas are $x = r\cos(\theta)$ and $y = r\sin(\theta)$. Substituting $r = 12\sin(\theta)$ into $y = r\sin(\theta)$, we get $y = 12\sin^2(\theta)$. Since $r = 12\sin(\theta)$, the maximum value of $r$ (and thus the radius) is 6 when $\sin(\theta) = 0.5$, which occurs at the top of the circle. The center is at (0, 6) because the circle is centered on the y-axis 6 units above the origin.

Find the Center and Radius of the Circle Whose Equation θ\thetaθ

Find the Center and Radius of the Circle Whose Equation $\theta$