Question

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

Options

• a = b

• a = −b

• ab = 1

• ab = 2

Answer 1

`ab = 1`

 

\[\text{ Given }: \]

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

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

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

\[\text { Now,} \]

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

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

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

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

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

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

\[\text { It is given that the curves cut orthogonally }.\]

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

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

\[ \Rightarrow ab = 1\]