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UPSC NDA II 2023 Mathematics Solved Paper

Question Numbers: 51-52

Consider the following for the next items that follow:

A parabola passes through (1, 2) and satisfies the differential equation

\frac{dy}{dx}=\frac{2y}{x},x>0,y>0.

Question : 51 of 120

Marks: +1, -0

What is the directrix of the parabola ?

Solution:

Concept:

The directrix of a parabola

x^2 = 4ay

is the line

y + a = 0

y = -a

Explanation:

We are given the differential equation

\frac{dy}{dx} = \frac{2y}{x}

with

x > 0, y > 0

Separate variables:

\frac{dy}{y} = \frac{2dx}{x}

Integrate both sides:

\int \frac{dy}{y} = \int \frac{2dx}{x}

gives

\ln y = 2\ln x + \ln c

Thus

\ln y = \ln(x^2 c)

, so

y = c x^2

The parabola passes through

(1,2)

2 = c \cdot 1^2 \Rightarrow c = 2

Hence the equation is

y = 2x^2

x^2 = \frac{y}{2}

Compare

x^2 = \frac{y}{2}

with the standard form

x^2 = 4ay

4a = \frac{1}{2} \Rightarrow a = \frac{1}{8}

The directrix is

y = -a = -\frac{1}{8}

Answer:

\boxed{y = -\frac{1}{8}}

(Option A).

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