Q220 / 270 · NDA 2023 (Phase 2) · Mathematics

What is the differential equation of all parabolas of the type $y^2=4 a(x-b)$ ?

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

Solution

Since, $y^2=4 a x-4 a b$ $\Rightarrow 2 y \frac{d y}{d x}=4 a \Rightarrow y \frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^2=0$

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

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