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NDA 2019
Exam
NDA
Year
2019
Subject
Mathematics
Chapter
Differential Equations

NDA 2019 · Mathematics · Differential Equations

The differential equation of the system of circles touching the y...

The differential equation of the system of circles touching the yy-axis at the origin is

Options

  1. Ax2+y2−2xydydx=0x^2+y^2-2 x y \frac{d y}{d x}=0
  2. Bx2+y2+2xydydx=0x^2+y^2+2 x y \frac{d y}{d x}=0
  3. Cx2−y2+2xydydx=0x^2-y^2+2 x y \frac{d y}{d x}=0
  4. Dx2−y2−2xydydx=0x^2-y^2-2 x y \frac{d y}{d x}=0

Correct answer

C. x2−y2+2xydydx=0x^2-y^2+2 x y \frac{d y}{d x}=0

Step-by-step solution

The equation of circle touching yy-axis at origin is (x−α)2+(y−0)2=α2⇒x2+α2−2αx+y2=α2⇒x2+y2−2αx=0⇒x+y2x−2α=0\begin{aligned} & (x-\alpha)^2+(y-0)^2=\alpha^2 \\ & \Rightarrow x^2+\alpha^2-2 \alpha x+y^2=\alpha^2 \\ & \Rightarrow x^2+y^2-2 \alpha x=0 \\ & \Rightarrow x+\frac{y^2}{x}-2 \alpha=0\end{aligned} Differentiating, 1+x⋅2y⋅dydx−y2x2=0⇒x2+2xy⋅dydx−y2=01+\frac{x \cdot 2 y \cdot \frac{d y}{d x}-y^2}{x^2}=0\Rightarrow x^2+2 x y \cdot \frac{d y}{d x}-y^2=0
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