Question

Differential equation of the family of parabolas whose vertex lie on the $x-$ axis and focus as origin is

• A. $y{\left(\frac{dy}{dx}\right)}^2+2x\frac{dy}{dx}-y=0$
• B. $y{\left(\frac{dy}{dx}\right)}^2+2x\frac{dy}{dx}+y=0$
• C. $y{\left(\frac{dy}{dx}\right)}^2+x\frac{dy}{dx}-y=0$
• D. $y{\left(\frac{dy}{dx}\right)}^2+x\frac{dy}{dx}+y=0$

Answer 1

The correct option is A $y{\left(\frac{dy}{dx}\right)}^2+2x\frac{dy}{dx}-y=0$

$\because$ focus and vertex both lie on the $x-$axis
$\therefore$ axis of the parabola will be $x-$axis
Let coordinates of the vertex is $(-a,0)$
Equation of the Directrix will be $x+2a=0$
Let any point on the parabola be $P(x,y)$
Now $PS=PM$
$P{S}^2=P{M}^2{x}^2+y^2=(x+2a{)}^2{x}^2+y^2=x^2+4a^2+4ax{y}^2=4a(a+x)$
Differentiating with respect to $x$
$2y\frac{dy}{dx}=4aa=\frac{y}{2}\cdot \frac{dy}{dx}\therefore y^2=2y\cdot \frac{dy}{dx}(\frac{y}{2}\cdot \frac{dy}{dx}+x)y{\left(\frac{dy}{dx}\right)}^2+2x\frac{dy}{dx}-y=0$