Q253 / 270 · NDA 2015 (Phase 2) · Mathematics

The differential equation of the family of circles passing through the origin and having centres on the x -axis is

  • A
    $2 x y \frac{d y}{d x}=x^2-y^2$
  • Correct answer
    B
    $2 x y \frac{d y}{d x}=y^2-x^2$
  • C
    $\quad 2 x y \frac{d y}{d x}=x^2+y^2$
  • D
    $2 x y \frac{d y}{d x}+x^2+y^2=0$

Solution

Eq. of family of circles passing through the origin \& having centres on the x -axis is: $\begin{aligned} & x^2+y^2+2 g x=0 \\ & 2 x+2 y \cdot \frac{d y}{d x}+2 g=0 \\ & \Rightarrow g=-\left[x+y \frac{d y}{d x}\right] \end{aligned}$ [on differentiating] Putting the value of (g) in eq. (1) we get; $2 x y \frac{d y}{d x}=y^2-x^2$

Hence, the correct answer is option: 'B'.

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