A variable tangent to the ellipse frac{x^2}{a | vedclass

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(D) The equation of the ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.The equation of the tangent at point $(a \cos 	heta, b \sin 	heta)$ is $\frac

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$\frac{a^2}{x^2}+\frac{b^2}{y^2}=4$

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(D) The equation of the ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.The equation of the tangent at point $(a \cos 	heta, b \sin 	heta)$ is $\frac{x \cos 	heta}{a}+\frac{y \sin 	heta}{b}=1$.The tangent meets the $x$-axis at $A\left(\frac{a}{\cos 	heta}, 0ight)$ and the $y$-axis at $B\left(0, \frac{b}{\sin 	heta}ight)$.Let $(x, y)$ be the midpoint of $AB$. Then:$x = \frac{a}{2 \cos 	heta} \Rightarrow \cos 	heta = \frac{a}{2x}$$y = \frac{b}{2 \sin 	heta} \Rightarrow \sin 	heta = \frac{b}{2y}$Using the identity $\sin^2 	heta + \cos^2 	heta = 1$,we get:$\left(\frac{b}{2y}ight)^2 + \left(\frac{a}{2x}ight)^2 = 1$$\frac{b^2}{4y^2} + \frac{a^2}{4x^2} = 1$$\frac{a^2}{x^2} + \frac{b^2}{y^2} = 4$.Thus,the locus of the midpoint is $\frac{a^2}{x^2} + \frac{b^2}{y^2} = 4$.

$A$ variable tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ makes intercepts on both the axes. The locus of the midpoint of the portion of the tangent between the coordinate axes is

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