The differential equation representing family of circles with centre at origin and radius r units is-
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \displaystyle\frac{dy}{dx}=-\frac{x}{y} \)
The family is \( x^2+y^2=r^2 \), with r an arbitrary constant. Differentiating with respect to x eliminates r:
\( \displaystyle 2x+2y\frac{dy}{dx}=0 \)
\( \displaystyle \frac{dy}{dx}=-\frac{x}{y} \)
Why the other options are incorrect
Option (2) \( \displaystyle\frac{dy}{dx}=\frac{2r}{y} \) — Incorrect. It still contains the constant r, so it does not represent the whole family.
Option (3) \( xdy=-ydx \) — Incorrect. \( x\,dy=-y\,dx \) gives \( \frac{dy}{dx}=-\frac{y}{x} \), the equation of the family xy = c, not of these circles.
Option (4) \( ydy=-2xdx \) — Incorrect. \( y\,dy=-2x\,dx \) integrates to \( y^2+2x^2=c \), a family of ellipses; the factor 2 is wrong.
CBSE Applied Mathematics (Class XII):
Formation of a Differential Equation
