Question

The line $ax + by = 1$ cuts ellipse $cx^2 + dy^2 = 1$ only once if

• A. $ca^2 + db^2 = 1 $
• B. $\frac{c}{a^{2}} + \frac{d}{b^{2}} = 1 $
• C. $\frac{a^{2}}{c} + \frac{b^{2}}{d} = 1 $
• D. $ac^2 + bd^2 = 1 $

Answer 1

Answer: (C)

Clearly $a x+b y=1$
i.e $y=-\frac{a}{b} x+\frac{1}{b}$ is tangent to
$c x^{2}+d y^{2}=1$
$\Rightarrow \frac{x^{2}}{\frac{1}{c}}+\frac{y^{2}}{\frac{1}{d}}=1$
$\therefore \left(\frac{1}{b}\right)^{2}=\left(\frac{1}{c}\right)\left(-\frac{a}{b}\right)^{2}+\left(\frac{1}{d}\right)$
$\Rightarrow 1=\frac{a^{2}}{c}+\frac{b^{2}}{d}$