Question

The differential equation of family of parobalas with foci at the origin and axis along the X- axis, is

• A. $y{\left(\frac{dy}{dx}\right)}^2+2x\frac{dy}{dx}-y=0$
• B. $y{\left(\frac{dy}{dx}\right)}^2+2y\frac{dy}{dx}-y=0$
• C. $y{\left(\frac{dy}{dx}\right)}^2+2x\frac{dy}{dx}+y=0$
• D. None of these

Answer 1

The correct option is A

$y{\left(\frac{dy}{dx}\right)}^2+2x\frac{dy}{dx}-y=0$

Let the directrix be $x=-2a$ and latus rectum be 4a then the equation of the parabola is (distance from focus = distance from directrix).
$x^2+y^2=(2a+x{)}^2\Rightarrow y^2=4a(a+x)$
On differentiating w.r.t. $x$, we get $y.\frac{dy}{dx}-2a\Rightarrow a=\frac{y}{2}\frac{dy}{dx}$
On putting this value of a in eq. (i) the differental equation is
$y^2=2y=\frac{dy}{dx}(\frac{y}{2}\frac{dy}{dx}+x)\Rightarrow y{\left(\frac{dy}{dx}\right)}^2+2x\left(\frac{dy}{dx}\right)-y=0$