Question

Find the points on r = a (1 + cosθ), a > 0 , where the tangent line is horizontal.

Answer 1

The tangent line is horizontal when max or min occur for "y"
normally set dy/dx = 0
since its in polar we can use dy/dtheta = 0 which will find the angles where max/min occur

\[y = r \sin \theta = a(1+\cos \theta) \sin \theta\]
\[\frac{dy}{d \theta} = a (-\sin^2 \theta +\cos \theta + \cos^2 \theta) = 0\]
\[\rightarrow 2 \cos^2 \theta + \cos \theta -1 = 0\]
\[\rightarrow (2 \cos \theta -1) (\cos \theta +1) = 0\]
\[\cos \theta = \frac{1}{2} \rightarrow \theta = \pm \frac{\pi}{3}\]
\[\cos \theta = -1 \rightarrow \theta = \pi\]