The equation r cos theta = 2 a sin^2 theta re | vedclass

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Question

(C) Given the polar equation: $r \cos 	heta = 2a \sin^2 	heta$. We know the conversion relations: $x = r \cos 	heta$, $y = r \sin 	heta$, and $r^2

Vedclass · Accepted Answer

$x^3 = y^2(2a - x)$

Vedclass · Answer

(C) Given the polar equation: $r \cos 	heta = 2a \sin^2 	heta$. We know the conversion relations: $x = r \cos 	heta$, $y = r \sin 	heta$, and $r^2 = x^2 + y^2$. From the given equation, we have $r \cos 	heta = 2a \sin^2 	heta$. Multiply both sides by $r^2$: $r^3 \cos 	heta = 2a (r \sin 	heta)^2$. Since $r^2 = x^2 + y^2$, we have $r = \sqrt{x^2 + y^2}$. Substituting $x = r \cos 	heta$ and $y = r \sin 	heta$: $r^2 (r \cos 	heta) = 2a (r \sin 	heta)^2$ $(x^2 + y^2)x = 2a y^2$. Rearranging the terms: $x(x^2 + y^2) = 2a y^2$ $x^3 + xy^2 = 2a y^2$ $x^3 = 2a y^2 - xy^2$ $x^3 = y^2(2a - x)$. Thus, the correct option is $C$.

The equation $r \cos \theta = 2 a \sin^2 \theta$ represents the curve

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