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Question
Write the angle between the curves y = e−x and y = ex at their point of intersections ?
Solution
\[\text { Given }: \]
\[y = e^{- x} . . . \left( 1 \right)\]
\[y = e^x . . . \left( 2 \right)\]
\[\text { On substituting the value of y in (1), we get }\]
\[ e^{- x} = e^x \]
\[ \Rightarrow x = 0\]
\[\text { and }\]
\[y = 1 ...................[\text { From } (2)]\]
\[\text { On differentiating (1) w.r.t.x,we get }\]
\[\frac{dy}{dx} = - e^{- x} \]
\[ \Rightarrow m_1 = \left( \frac{dy}{dx} \right)_\left( 0, 1 \right) = - 1\]
\[\text { On differentiating (2) w.r.t.x,we get }\]
\[\frac{dy}{dx} = e^x \]
\[ \Rightarrow m_2 = \left( \frac{dy}{dx} \right)_\left( 0, 1 \right) = 1\]
\[ \because m_1 \times m_2 = - 1\]
Since the multiplication of the slopes is - 1.
So the slopes are perpendicular to each other.
\[ \therefore \text { Required angle } = \frac{\pi}{2}\]
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