Practicing Success
The solution of the equation $(2+x) d y-y d x=0$ represents a curve passing through a fixed point $P$, then the area of equilateral triangle with $P$ as one vertex and $x+y=0$ as its one side, is |
$2 \sqrt{3}$ $\sqrt{3}$ $\frac{2}{\sqrt{3}}$ $\frac{4}{\sqrt{3}}$ |
$\frac{2}{\sqrt{3}}$ |
We have, $(2+x) d y-y d x=0 \Rightarrow \frac{1}{y} d y-\frac{1}{2+x} d x=0$ On integrating, we obtain $\log y-\log (x+2)=\log C \Rightarrow y=C(x+2)$ Clearly, it represents a curve passing through a fixed point $P(-2,0)$. The altitude of the equilateral triangle having one vertex at $P(-2,0)$ and opposite side $x+y=0$ is $p=\left|\frac{-2+0}{\sqrt{1+1}}\right|=\sqrt{2}$ So, its area is $\frac{p^2}{\sqrt{3}}=\frac{2}{\sqrt{3}}$ sq. units |