Mathematics

Problem

The astroid is represented parametrically by

x = a\cos^3 \theta, \quad y = a\sin^3 \theta, \quad 0 < \theta < 2\pi, \; a>0.

Work through the tasks below. (a) Show that the slope of the tangent at any point is given by

\dfrac{dy}{dx} = -\tan \theta

. (b) Determine the coordinates of all points on the curve where the tangent is (i) horizontal, and (ii) vertical. (c) Find the equation of the tangent to the astroid at

\theta = \dfrac{\pi}{6}.

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