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# The locus of the mid- points of the portion of the tangents to the ellipse intercepted between the axis is :

$\begin{array}{1 1}(A)\;a^2y^2+b^2x^2=4x^2y^2 \\(B)\;a^2x^2+b^2y^2=4x^2y^2\\(C)\;x^2+y^2=a^2 \\(D)\;x^2+y^2=b^2 \end{array}$

Any tangent to the ellipse is $\large\frac{x}{a}$$\cos \theta + \large\frac{y}{b}$$ \sin \theta=1$
It meets the axes at $A \bigg( \large\frac{a}{\cos \theta}$$, 0\bigg ),B \bigg( 0, \large\frac{b}{\sin \theta} \bigg) If (h,k) be the mid point of AB then, 2h =\large\frac{a}{\cos \theta} 2k =\large\frac{b}{\sin \theta} \therefore \cos \theta=\large\frac{a}{2h},$$ \sin \theta=\large\frac{b}{2k}$
$\therefore \cos^2 \theta + \sin ^2 \theta =1$