Question

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

• A. 0
• B. 1
• C. $-1$
• D. 2

Answer 1

Answer: (B)

Let the equation of tangent at the point $\left(x_1,y_1\right)$ 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 $\left(\frac{a^2}{x_1},0\right)$ and $\left(0,\frac{b^2}{y_1}\right)$
but $\frac{a^2}{x_1}=h$ and $\frac{b^2}{y_1}=k$ (given)
$\Rightarrow x_1=\frac{a^2}{h},y_1=\frac{b^2}{k}$
$\Rightarrow \frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}=1$
$\therefore \frac{a^2}{h^2}+\frac{b^2}{k^2}=1$