Question:
The curves $y=a e^{x}$ and $y=b e^{-x}$ cut orthogonally, if
A. $a=b$
B. $a=-b$
C. $a b=1$
D. $a b=2$
Solution:
Given that the curves $y=a e^{x}$ and $y=b e^{-x}$
Differentiating both of them w.r.t. $x$,
$\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{ae}^{\mathrm{x}}$ and $\frac{\mathrm{dy}}{\mathrm{dx}}=-\mathrm{be}^{-\mathrm{x}}$
Let $m_{1}=a e^{x}$ and $m_{2}=-b e^{-x}$
$\mathrm{m}_{1} \times \mathrm{m}_{2}=-1$
(Because curves cut each other orthogonally)
$\Rightarrow-a b=-1$
$\Rightarrow a b=1$