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Question
The curves y = aex and y = be−x cut orthogonally, if ___________ .
Options
a = b
a = −b
ab = 1
ab = 2
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Solution
`ab = 1`
\[\text{ Given }: \]
\[y = a e^x . . . \left( 1 \right)\]
\[y = b e^{- x} . . . \left( 2 \right)\]
\[\text { Let the point of intersection of these two curves be }\left( x_1 , y_1 \right).\]
\[\text { Now,} \]
\[\text { On differentiating (1) w.r.t.x, we get }\]
\[\frac{dy}{dx} = a e^x \]
\[ \Rightarrow m_1 = \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) = a e^{x_1} \]
\[\text { Again, on differentiating (2) w.r.t.x, we get }\]
\[\frac{dy}{dx} = - b e^{- x} \]
\[ \Rightarrow m_2 = \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) = - b e^{- x_1} \]
\[\text { It is given that the curves cut orthogonally }.\]
\[ \therefore m_1 \times m_2 = - 1\]
\[ \Rightarrow a e^{x_1} \times \left( - b e^{- x_1} \right) = - 1\]
\[ \Rightarrow ab = 1\]
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