The portion of the tangent to the curve x^{2/ | vedclass

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(C) Given the curve $x^{2/3} + y^{2/3} = a^{2/3}$.Let the parametric coordinates of any point on the curve be $(x, y) = (a \cos^3 	heta, a \sin^3 	h

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constant

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(C) Given the curve $x^{2/3} + y^{2/3} = a^{2/3}$.Let the parametric coordinates of any point on the curve be $(x, y) = (a \cos^3 	heta, a \sin^3 	heta)$.The slope of the tangent $\frac{dy}{dx}$ is given by $\frac{dy/d	heta}{dx/d	heta}$.$\frac{dy}{d	heta} = 3a \sin^2 	heta \cos 	heta$ and $\frac{dx}{d	heta} = -3a \cos^2 	heta \sin 	heta$.Thus, $\frac{dy}{dx} = \frac{3a \sin^2 	heta \cos 	heta}{-3a \cos^2 	heta \sin 	heta} = -	an 	heta$.The equation of the tangent at $(a \cos^3 	heta, a \sin^3 	heta)$ is $y - a \sin^3 	heta = -	an 	heta (x - a \cos^3 	heta)$.Simplifying, $y \cos 	heta - a \sin^3 	heta \cos 	heta = -x \sin 	heta + a \cos^3 	heta \sin 	heta$.$x \sin 	heta + y \cos 	heta = a \sin 	heta \cos 	heta (\sin^2 	heta + \cos^2 	heta) = a \sin 	heta \cos 	heta$.Dividing by $a \sin 	heta \cos 	heta$, we get $\frac{x}{a \cos 	heta} + \frac{y}{a \sin 	heta} = 1$.The $x$-intercept is $a \cos 	heta$ and the $y$-intercept is $a \sin 	heta$.The length of the intercepted portion is $\sqrt{(a \cos 	heta)^2 + (a \sin 	heta)^2} = \sqrt{a^2(\cos^2 	heta + \sin^2 	heta)} = a$.Since $a$ is a constant, the length of the intercepted portion is constant.

The portion of the tangent to the curve $x^{2/3} + y^{2/3} = a^{2/3}, a > 0$ at any point, intercepted between the axes, is:

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