Question

The locus of the mid-point of the portion of a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ included between the coordinate axes is

• A. $\frac{a^2}{x^2}+\frac{b^2}{y^2}=1$
• B. $\frac{a}{x^2}+\frac{b}{y^2}=2$
• C. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=4$
• D. $\frac{a}{x^2}+\frac{b}{y^2}=4$

Answer 1

The correct option is D $\frac{a}{x^2}+\frac{b}{y^2}=4$

Let any tangent of ellipse is

$\frac{xcos\theta }{a}+\frac{ysin\theta }{b}=1$

Let it meets axes at $A(\frac{a}{cos\theta },0)andB(0,\frac{b}{sin\theta })$

Let midpoint of AB is (h,k) then

$2h=\frac{a}{cos\theta },2k=\frac{b}{sin\theta }$

$co{s}^2\theta +si{n}^2\theta =1$

$⟹\frac{a^2}{4h^2}+\frac{b^2}{4k^2}=1$

$⟹a{k}^2+b{h}^2=4h^2{k}^2$

Hence the locus is $a{y}^2+b{x}^2=4x^2{y}^2$

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