If the curves


If the curves $y=2 e^{x}$ and $y=a e^{-x}$ interest orthogonally, then $a=$

A. $\frac{1}{2}$

B. $-\frac{1}{2}$

C. 2

D. $2 e^{2}$


Given that the curves $y=2 e^{x}$ and $y=a e^{-x}$

Differentiating both of them w.r.t. $x$,

$\frac{\mathrm{dy}}{\mathrm{dx}}=2 \mathrm{e}^{\mathrm{x}}$ and $\frac{\mathrm{dy}}{\mathrm{dx}}=-\mathrm{ae}^{-\mathrm{x}}$

Let $\mathrm{m}_{1}=2 \mathrm{e}^{\mathrm{x}}$ and $\mathrm{m}_{2}=-\mathrm{ae}^{-\mathrm{x}}$

$m_{1} \times m_{2}=-1$

(Because curves cut each other orthogonally)

$\Rightarrow-2 a=-1$

$\Rightarrow a=\frac{1}{2}$

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