Let . A polar curve is given by . In this problem you will translate the curve to…
MATHEMATICS · Multi-part · AP · AP Precalculus · Trigonometric and Polar Functions · Trigonometric and Polar Functions
Problem
Let a>0. A polar curve is given by r=2acosθ. In this problem you will translate the curve to Cartesian form and analyze its geometric features.
(a) Starting from x=rcosθ and y=rsinθ, derive a Cartesian equation that is equivalent to r=2acosθ. Show every algebraic step, including the elimination of θ.
(b) Verify directly—by substituting x and y back into your equation from Part (a)—that the point corresponding to θ=0 indeed lies on the Cartesian curve. State the Cartesian coordinates of this point.
(c) Identify the center and radius of the curve obtained in Part (a). Explain how these characteristics can be read from the Cartesian form you derived.