Question

If the tangent line to an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ cuts intercepts $h$ and $k$, then $\frac{a^2}{h^2}+\frac{b^2}{k^2}=$

• A. $0$
• B. $1$
• C. $-1$
• D. $2$

Answer 1

Answer: (B)

$1$

Let the equation of tangent at the point $(x_1,y_1)$ be

$\frac{x{x}_1}{a^2}+\frac{y{y}_1}{b^2}=1\Rightarrow \frac{x}{\frac{a^2}{x_1}}+\frac{y}{\frac{b^2}{y_1}}=1$

Which meets axes at $(\frac{a^2}{x_1},0)$ and $(0,\frac{b^2}{y_1})$

But $\frac{a^2}{x_1}=h$ and $\frac{b^2}{y_1}=k$ (given)

$\Rightarrow x_1=\frac{a^2}{h},y_1=\frac{b^2}{k}\Rightarrow \frac{{x_1}^2}{a^2}+\frac{{y_1}^2}{b^2}=1$

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