Question

The families of curves represented by the solution of the equation
$\frac{dy}{dx}=-{\left(\frac{x}{y}\right)}^n$
For $n=-1$ and $n=+1$, respectively, are

• A. Hyperbola and Parabolas
• B. Circles and Hyperbolas
• C. Parabolas and Circles
• D. Hyperbolas and Circles

Answer 1

The correct option is D Hyperbolas and Circles

$\frac{dy}{dx}=-{\left(\frac{x}{y}\right)}^n$
for$n=-1$
$\frac{dy}{dx}=-{\left(\frac{x}{y}\right)}^{-1}$
$\int \frac{dy}{y}=-\int \frac{dx}{x}$
$lny=-ln\left(x\right)+lnk$
$ln\left(y\right)+ln\left(x\right)=ln\left(k\right)$
$ln\left(xy\right)=ln\left(k\right)$
taking log inverse
$xy=k$ Rectangular hyperbola
for $n=1$
$\frac{dy}{dx}=-\left(\frac{x}{y}\right)$
$\int y.dy=-\int x.dx$
$\frac{y^2}{2}=-\frac{x^2}{2}+c$
$x^2+y^2=c$ circle.
for $n=-1;$ Hyperbolas
for $n=+;$ circles