The differential equation of all parabolas, whose axes are parallel to $\mathrm{Y}$-axis, is

The differential equation of all parabolas, whose axes are parallel to $\mathrm{Y}$-axis, is
  1. $y_3=1$
  2. $y_3=-1$
  3. $y_3=0$
  4. $yy_3+y_1=0$

Solution

Parabola whose axes are parallel to $\mathrm{Y}$-axis. Vertex is not $(0,0)$ Equation becomes $(x-\mathrm{h})^2=4 \mathrm{~b}(y-\mathrm{k})$ Differentiating w.r.t. $x$, we get $2(x-h)=4 b\left(\frac{d y}{d x}\right)$ Again differentiating w.r.t. $x$, we get $2=4 \mathrm{~b}\left(\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}\right)$ Again differentiating w.r.t. $x$, we get $\begin{aligned} & 0=4 b\left(\frac{\mathrm{d}^3 y}{\mathrm{~d} x^3}\right) \\ \therefore \quad & \frac{\mathrm{d}^3 y}{\mathrm{~d} x^3}=0 \\ & \text { i.e., } y_3=0 \end{aligned}$

Asked in: MHT CET 2023 (09 May Shift 1)

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