The solution of the differential equation \( \displaystyle\frac{dy}{dx}=\frac{x}{y} \) represents a family of
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → hyperbolas
The variables separate:
\( \displaystyle y\,dy=x\,dx \)
Integrating both sides,
\( \displaystyle \frac{y^2}{2}=\frac{x^2}{2}+C_1 \)
\( \displaystyle y^2-x^2=C\quad(C=2C_1) \)
The squares of x and y appear with opposite signs, so for C ≠ 0 each curve is a rectangular hyperbola. The equation therefore represents a family of hyperbolas.
Why the other options are incorrect
Option (1) circles — Incorrect. A circle needs \( x^2+y^2=C \), with both squares of the same sign; that comes from \( \frac{dy}{dx}=-\frac{x}{y} \).
Option (2) ellipses — Incorrect. An ellipse also needs \( x^2 \) and \( y^2 \) with the same sign (and different coefficients), which does not happen here.
Option (3) parabolas — Incorrect. A parabola has only one variable squared; here both x and y appear squared.