Question

If the curves y = 2 e^x and y = ae^−x intersect orthogonally, then a = _____________ .

Options

• 1/2

• −1/2

• 2

• 2e²

Answer 1

1/2

 

\[\text { Given }: \]

\[y = 2 e^x . . . \left( 1 \right)\]

\[y = a e^{- x} . . . \left( 2 \right)\]

\[\text { Let the point of intersection of these curves be }\left( x_1 , y_1 \right).\]

\[\text { On differentiating (1) w.r.t.x, we get }\]

\[\frac{dy}{dx} = 2 e^x \]

\[ \Rightarrow m_1 = \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) = 2 e^{x_1} \]

\[\text { Now, on differentiating (2) w.r.t.x, we get }\]

\[\frac{dy}{dx} = - a e^{- x} \]

\[ \Rightarrow m_2 = \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) = - a e^{- x_1} \]

\[\text { It is given that the curves are orthogonal }.\]

\[ \therefore m_1 \times m_2 = - 1\]

\[ \Rightarrow 2 e^{x_1} \times \left( - a e^{- x_1} \right) = - 1\]

\[ \Rightarrow 2a = 1\]

\[ \Rightarrow a = \frac{1}{2}\]