The families of curves represented by the solution of the equation \frac{d y}{d x}=-(\frac{x}{y})^{n} for {n}=-1 and {n}=+1, respectively, are

Correct Answer is : Hyperbolas and Circles

GATE Electronics and Communications (EC) 2019 Solved Paper

Question : 16 of 65

Marks: +1, -0

The families of curves represented by the solution of the equation

\;{d y}/{d x}=-(\;{x}/{y})^{n}

for

{n}=-1

and

{n}=+1

, respectively, are

Solution:

\;{d y}/{d x}=-(\;{x}/{y})^{n}

When

n=-1

\;{d y}/{d x}=-\;{x}/{y}

∴ \;{d y}/{y}=-\;{d x}/{x}

∴ \ln y=-\ln (x)+\ln (c)

∴ \ln (x y)=\ln (c)

∴ x y=c

This represents rectangular hyperbola.
Now for

n=+1

\;{d y}/{d x}=-\;{x}/{y}

∴ y d y=-x d x

∴ \;{y^{2}}/{2}=\;{-x^{2}}/{2}+c

∴ x^{2}+y^{2}=2 c

This represents family of circles.

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