# Find the locus of mid-points of the portions of the tangents to the ellipse $\large\frac{x^2}{a^2}+\frac{y^2}{b^2}$$=1 intercepted between the axis? \begin{array}{1 1}(a)\;a^2y^2+b^2x^2=4x^2y^2\\(b)\;a^2+b^2=4x^2y^2\\(c)\;a^2y^2+b^2x^2=2x^2y^2\\(d)\;\text{None of these}\end{array} ## 1 Answer Let P(x_1,y_1) be any point on the ellipse \large\frac{x^2}{a^2}+\frac{y^2}{b^2}$$=1$-----(1)
Equation of tangent at $(x_1,y_1)$ is
$\large\frac{xx_1}{a^2}+\frac{yy_1}{b^2}$$=1 This meet the co-ordinate axis at Q(\large\frac{a^2}{x_1},$$0)$ and $R(0,\large\frac{b^2}{y_1})$
Let $M(h,k)$ be the mid-point of QR then
$h=\large\frac{\Large\frac{a^2}{x_1}+0}{2},$$k=\large\frac{0+\Large\frac{b^2}{y_1}}{2} x_1=\large\frac{a^2}{2h},y_1=\large\frac{b^2}{2k} Since (x_1,y_1) lie on equ(1) hence \large\frac{(\Large\frac{a^2}{2h})^2}{a^2}+\frac{(\Large\frac{b^2}{2k})^2}{b^2}$$=1$