Question

If the curve ay + x² = 7 and x³ = y cut orthogonally at (1, 1), then a is equal to _____________ .

Options

• 1

• `-6`

• 6

• 0

Answer 1

`6`

 

\[\text { Given }: \]

\[ay + x^2 = 7 . . . \left( 1 \right)\]

\[ x^3 = y . . . \left( 2 \right)\]

\[\text { Point }=\left( 1, 1 \right)\]

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

\[a\frac{dy}{dx} + 2x = 0\]

\[ \Rightarrow \frac{dy}{dx} = \frac{- 2x}{a}\]

\[ \Rightarrow m_1 = \left( \frac{dy}{dx} \right)_\left( 1, 1 \right) = \frac{- 2}{a}\]

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

\[3 x^2 = \frac{dy}{dx}\]

\[ \Rightarrow m_2 = \left( \frac{dy}{dx} \right)_\left( 1, 1 \right) = 3\]

\[\text { It is giventhat the curves are orthogonal at the given point }.\]

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

\[ \Rightarrow \frac{- 2}{a} \times 3 = - 1\]

\[ \Rightarrow a = 6\]