Q202 / 270 · NDA 2023 (Phase 2) · Mathematics

A parabola pases through $(1,2)$ and satisfies the differential equation $\frac{d y}{d x}=\frac{2 y}{x}, x \gt 0, y \gt 0$. What is the length of latus rectum of the parabola?

  • A
    1
  • Correct answer
    B
    $\frac{1}{2}$
  • C
    $\frac{1}{4}$
  • D
    $\frac{1}{8}$

Solution

$\begin{aligned} & \text { Since, } \frac{d y}{d x}=\frac{2 y}{x} \\ & \int \frac{d y}{y}=\int \frac{2 d x}{x} \Rightarrow \ln y=2 \ln x+\ln c \Rightarrow y=x^2 c \\ & \text {it satisfy }(1,2) \\ & \quad \Rightarrow 2=1 C \Rightarrow C=2 \Rightarrow y=2 x^2\end{aligned}$ Since, length of lotus Rectum of Parabola $=4 \times \frac{1}{8}=\frac{1}{2}$

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

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