JEE Mains · Maths · STD 12 - 9. differential equations
If the solution curve \(y=y(x)\) of the differential equation \(y ^{2} dx +\left( x ^{2}- xy + y ^{2}\right) dy =0\), which passes through the point \((1,1)\) and intersects the line \(y=\sqrt{3} x\) at the point \((\alpha, \sqrt{3} \alpha)\), then value of \(\log _{e}(\sqrt{3} \alpha)\) is equal to
- A \(\frac{\pi}{3}\)
- B \(\frac{\pi}{2}\)
- C \(\frac{\pi}{12}\)
- D \(\frac{\pi}{6}\)
Answer & Solution
Correct Answer
(C) \(\frac{\pi}{12}\)
Step-by-step Solution
Detailed explanation
\(y^{2} d x-x y d y=-\left(x^{2}+y^{2}\right) d y\) \(y(y d x-x d y)=-\left(x^{2}+y^{2}\right) d y\) \(-y(x d x-y d x)=-\left(x^{2}+y^{2}\right) d y\) \(\frac{x d y-y d x}{x^{2}}=\left(1+\frac{y^{2}}{x^{2}}\right) \frac{d y}{y}\)…
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