Question

Find the equations of tangents to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ which make equal intercepts on the coordinates axes.

Answer 1

Given the equation of the ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.....(1).

We have the equation of the tangent at $(h,k)$ to the ellipse is ,

$\frac{xh}{a^2}+\frac{yk}{b^2}=1$

or, $\frac{x}{\frac{a^2}{h}}+\frac{y}{\frac{b^2}{k}}=1$

According to the problem,

$\frac{a^2}{h}=\frac{b^2}{k}$ [ Since the intercepts on the co-ordinate axes are same]

or, $h=\frac{k{a}^2}{b^2}$......(2).

We also have, $(h,k)$ lies on (1),

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

or, $\frac{k^2{a}^2}{b^4}+\frac{k^2}{b^2}=1$ [ Using (2)]

or, $k^2=\frac{b^4}{a^2+b^2}$

or, $k=\pm \frac{b^2}{\sqrt{a^2+b^2}}$......(3).

Using value of $k$ in (2) we get, $h=\pm \frac{a^2}{\sqrt{a^2+b^2}}$.

So the equation of the tangent is $x+y=\pm \sqrt{a^2+b^2}$.