Question

If a curve passing through $P(4,2)$ and satisfies the differential equation $\frac{dy}{dx}=\frac{y}{x-y^2}$ is a conic then which of the following is correct

• A. focus of conic lies on line $y=2$
• B. equation of diretrix of conic is $4x-17=0$
• C. eccentricity of conic is $1$
• D. eccemtricity of conic is $3$

Answer 1

Answer: (A), (C)

The correct options are
A focus of conic lies on line $y=2$
C eccentricity of conic is $1$

$\frac{dx}{dy}=\frac{x-y^2}{y}=\frac{x}{y}-y$

$\frac{dx}{dy}+x\left(\frac{-1}{y}\right)=-y$

$x.If=\int -y.Ifdy+C$

If $=e^{\int \frac{1}{y}}=e^{-lny}=\frac{1}{y}$

$x.\frac{1}{y}=-\int y.\frac{1}{y}dy+C$

$\frac{x}{y}=-y+C$ $(4,2)$

$\frac{4}{2}=-2+C\Rightarrow C=4$

$\frac{x}{y}=-y+4\Rightarrow x=-y^2+4y$

$y^2-4y+4=x+4$

$y+(y-2{)}^2=(x+1)$

$e=1,axisy=2$