Q78 / 150 · MHT CET 2021 (23 Sep Shift 1) · Mathematics

The differential equation of the family of parabolas with focus at the origin and the $\mathrm{X}$ - axis as axis, is

  • Correct answer
    A
    $-y\left(\frac{d y}{d x}\right)^2=2 x \frac{d y}{d x}-y$
  • B
    $y\left(\frac{d y}{d x}\right)^2+2 x y \frac{d y}{d x}+y=0$
  • C
    $y\left(\frac{d y}{d x}\right)^2+4 x \frac{d y}{d x}=4 x y$
  • D
    $y\left(\frac{d y}{d x}\right)^2+y=2 x y \frac{d y}{d x}$

Solution

Equation of parabola is $y^2=4 a(x+a)$ $$ 2 y \frac{d y}{d x}=4 a \Rightarrow a=\frac{y \frac{d y}{d x}}{2} $$ Substituting value of ' $a$ ', we get $$ \begin{aligned} & y^2=4\left(\frac{y \frac{d y}{d x}}{2}\right)\left[x+\left(\frac{y \frac{d y}{d x}}{2}\right)\right]=\left(2 y \frac{d y}{d x}\right)\left(\frac{2 x+y \frac{d y}{d x}}{2}\right) \\ & 2 y^2=\left(2 y \frac{d y}{d x}\right)\left(2 x+y \frac{d y}{d x}\right)=4 x y \frac{d y}{d x}+2 y^2\left(\frac{d y}{d x}\right)^2 \\ & \therefore \quad y=2 x \frac{d y}{d x}+y\left(\frac{d y}{d x}\right)^2 \end{aligned} $$

Hence, the correct answer is option: 'A'.

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