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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Differential Equations

NDA 2023 · Mathematics · Differential Equations

A parabola pases through (1,2) and satisfies the differential...

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

Options

  1. A1
  2. B12\frac{1}{2}
  3. C14\frac{1}{4}
  4. D18\frac{1}{8}

Correct answer

B. 12\frac{1}{2}

Step-by-step solution

 Since, dydx=2yx∫dyy=∫2dxx⇒ln⁡y=2ln⁡x+ln⁡c⇒y=x2cit satisfy (1,2)⇒2=1C⇒C=2⇒y=2x2\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 =4×18=12=4 \times \frac{1}{8}=\frac{1}{2}
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