A particle is moving with a velocity $\vec{v}=K(y\hat{i}+x\hat{j})$, where K is a constant. The general equation for its path is:
(1) xy = constant
(2) y2 = x2 + constant
(3) y = x2 + constant
(4) y2 = x + constant
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Ans: (2) y2 = x2 + constant
Sol: ${dx\over dt}=ky, {dy\over dt}=kx$
Now, ${dy\over dx}={{dy\over dt}\over{dx\over dt}}={x\over y}$
$\implies ydy=xdx$
Integrating both side
$y^2=x^2+c$
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