# The locus of the mid -point of the portion of the tangents to the ellipse intercepted between the axes 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)$$B\bigg( 0, \large\frac{b}{\sin \theta} \bigg) If (h,k) be the mid - point of AB then 2h=\large\frac{a}{\cos \theta} 2k=\large\frac{b}{\sin \theta} \therefore \cos \theta=\large\frac{a}{2h} \qquad$$\sin \theta =\large\frac{b}{2k}$
$\therefore \cos ^2 \theta +\sin ^2 \theta=1$