Question

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

• A. Focus of the conic lies on line $y=2$.
• B. Equation of directrix of conic is $4x–17=0$.
• C. Eccentricity of the conic is $1$.
• D. Eccentricity of the conic is $3$.

Answer 1

Answer: (A), (B), (C)

The correct options are
A Focus of the conic lies on line $y=2$.
B Equation of directrix of conic is $4x–17=0$.
C Eccentricity of the conic is $1$.

$\frac{dx}{dy}=\frac{x}{y}-y$
$\Rightarrow \frac{dx}{dy}-\frac{1}{y}x=-y$
$IF=e^{\int -\frac{1}{y}dy}=\frac{1}{y}$
$\Rightarrow x\cdot \frac{1}{y}=-\int \frac{1}{y}\cdot ydy+c$
$\Rightarrow x\cdot \frac{1}{y}=-y+c$
$\Rightarrow x=-y^2+cy$
As the curve passes through $(4,2),$ we get $c=4$
Hence, the curve is $x=-y^2+4y$
$(x-4)=-(y-2{)}^2$
$\Rightarrow (y-2{)}^2=-(x-4).$
Eccentricity of parabola is always $1$