Question

Write the angle between the curves y = e^−x and y = e^x at their point of intersections ?

Answer 1

\[\text { Given }: \]

\[y = e^{- x} . . . \left( 1 \right)\]

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

\[\text { On substituting the value of y in (1), we get }\]

\[ e^{- x} = e^x \]

\[ \Rightarrow x = 0\]

\[\text { and }\]

\[y = 1 ...................[\text { From } (2)]\]

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

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

\[ \Rightarrow m_1 = \left( \frac{dy}{dx} \right)_\left( 0, 1 \right) = - 1\]

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

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

\[ \Rightarrow m_2 = \left( \frac{dy}{dx} \right)_\left( 0, 1 \right) = 1\]

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

Since the multiplication of the slopes is - 1.

So the slopes are perpendicular to each other.

\[ \therefore \text { Required angle } = \frac{\pi}{2}\]