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MathematicsSingle correct+4 / -1Question 18 of 90

JEE Main 2022 · 25 Jun · Shift 1, Mathematics question 18

If the solution curve y=y(x)y=y(x) of the differential equation y2dx+(x2−xy+y2)dy=0y^2dx+\left(x^2-xy+y^2\right)dy=0, which passes through the point (1,1)(1,1) and intersects the line y=3xy=\sqrt3x at the point (α,3α)\left(\alpha,\sqrt3\alpha\right), then value of log⁡e(3α)\log_e\left(\sqrt3\alpha\right) is equal to :

Choose one option
Answer and worked solution

Official answer: C

Homogeneous in xx. dxdy=−x2−xy+y2y2\dfrac{dx}{dy}=-\dfrac{x^2-xy+y^2}{y^2}; put x=vyx=vy: ydvdy=−(1+v2)y\dfrac{dv}{dy}=-(1+v^2).

Integrate. tan⁡−1v=−log⁡ey+C\tan^{-1}v=-\log_ey+C; at (1,1)(1,1), C=π4C=\frac\pi4.

On y=3xy=\sqrt3x. v=13v=\dfrac1{\sqrt3}: π6=−log⁡ey+π4\dfrac\pi6=-\log_ey+\dfrac\pi4, so log⁡e(3α)=log⁡ey=π12\log_e\left(\sqrt3\alpha\right)=\log_ey=\dfrac\pi{12}.

(C)

Topics:Homogeneous equations
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