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प्रश्न
If the curve ay + x2 = 7 and x3 = y cut orthogonally at (1, 1), then a is equal to _____________ .
पर्याय
1
`-6`
6
0
उत्तर
`6`
\[\text { Given }: \]
\[ay + x^2 = 7 . . . \left( 1 \right)\]
\[ x^3 = y . . . \left( 2 \right)\]
\[\text { Point }=\left( 1, 1 \right)\]
\[\text { On differentiating (1) w.r.t.x, we get }\]
\[a\frac{dy}{dx} + 2x = 0\]
\[ \Rightarrow \frac{dy}{dx} = \frac{- 2x}{a}\]
\[ \Rightarrow m_1 = \left( \frac{dy}{dx} \right)_\left( 1, 1 \right) = \frac{- 2}{a}\]
\[\text { Again, on differentiating (2) w.r.t.x, we get }\]
\[3 x^2 = \frac{dy}{dx}\]
\[ \Rightarrow m_2 = \left( \frac{dy}{dx} \right)_\left( 1, 1 \right) = 3\]
\[\text { It is giventhat the curves are orthogonal at the given point }.\]
\[ \therefore m_1 \times m_2 = - 1\]
\[ \Rightarrow \frac{- 2}{a} \times 3 = - 1\]
\[ \Rightarrow a = 6\]
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