Question

The curves y = ae^x and y = be^−x cut orthogonally, if

(a) a = b
(b) a = −b
(c) ab = 1
(d) ab = 2

Answer 1

(c) ab = 1

$$\begin{gathered} \mathrm{Given}: \\ y=a{e}^x...\left(1\right) \\ y=b{e}^{-x}...\left(2\right) \\ \text{Let the point of intersection of these two curves be}\left(x_1,y_1\right)\text{.} \\ \mathrm{Now}, \\ \mathrm{On}\mathrm{d}\text{ifferentiating (1) w.r.t.}\mathit{\text{x}}\text{, we get} \\ \frac{dy}{dx}=a{e}^x \\ \Rightarrow m_1={\left(\frac{dy}{dx}\right)}_{\left(x_1,y_1\right)}=a{e}^{x_1} \\ \mathrm{Again},\mathrm{on}\mathrm{d}\text{ifferentiating (2) w.r.t.}\mathit{\text{x}}\text{, we get} \\ \frac{dy}{dx}=-b{e}^{-x} \\ \Rightarrow m_2={\left(\frac{dy}{dx}\right)}_{\left(x_1,y_1\right)}=-b{e}^{-x_1} \\ \text{It is given that the curves cut orthogonally.} \\ \therefore m_1\times m_2=-1 \\ \Rightarrow a{e}^{x_1}\times \left(-b{e}^{-x_1}\right)=-1 \\ \Rightarrow ab=1 \end{gathered}$$