Question

If any tangent to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ intercepts lengths h and k on the axes, then

• A. $\frac{a^{2}}{h^{2}} - \frac{b^{2}}{k^{2}} = 1$
• B. $\frac{a^{2}}{h^{2}} + \frac{b^{2}}{k^{2}} = 1$
• C. $\frac{a^{2}}{k^{2}} - \frac{b^{2}}{h^{2}} = 1$
• D. $\frac{a^{2}}{k^{2}} + \frac{b^{2}}{h^{2}} = 1$

Answer 1

Answer: (B)

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

Answer 2

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

$\frac{xx_{1}}{a^{2}} + \frac{yy_{1}}{b^{2}} = 1$ .... (2)

The tangent at P meets x-axis i.e. y = 0 where

$\frac{xx_{1}}{a^{2}} = 1orx = \frac{a^{2}}{x_{1}}$

But the intercept on x-axis is given to be h

∴ $\frac{a^{2}}{x_{1}} = horx_{1} = \frac{a^{2}}{h}$The tangent at P meets y-axis i.e. x = 0 where $\frac{yy_{1}}{b^{2}} = 1or\frac{b^{2}}{y_{1}}$.

But the intercept on y-axis is given to be k.

∴ $\frac{b^{2}}{y_{1}} = kory_{1} = \frac{b^{2}}{k}$Since P lies on (1),

∴ $\frac{x_{1}^{2}}{a^{2}} + \frac{y_{1}^{2}}{b^{2}} = 1$

or$\frac{a^{4}/h^{2}}{a^{2}} + \frac{b^{4}/k^{2}}{b^{2}} = 1$ [Using (3) and (4)]

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