The curves y = aex and y = be−x cut orthogonally, if ___________ .

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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 } dy/dx = a e^x \Rightarrow m_1 = \left( dy/dx \right)_\left( x_1 , y_1 \right) = a e^{x_1} \text { Again, on differentiating (2) w.r.t.x, we get } dy/dx = - b e^{- x} \Rightarrow m_2 = \left( 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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