The astroid is represented parametrically by x = a ^3

MATHEMATICS · Subjective · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

The astroid is represented parametrically by x=acos⁡3θ,y=asin⁡3θ,0<θ<2π,  a>0.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 dydx=−tan⁡θ\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 θ=π6.\theta = \dfrac{\pi}{6}.

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