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NDA 2017
Exam
NDA
Year
2017
Subject
Mathematics
Chapter
Differential Equations

NDA 2017 · Mathematics · Differential Equations

The general solution of (d y)/(d x)=(a x+h)/(b y+k) represents a...

The general solution of dydx=ax+hby+krepresentsacircleonlywhen\frac{d y}{d x}=\frac{a x+h}{b y+k} represents a circle only when

Options

  1. Aa=b=0\mathrm{a}=\mathrm{b}=0
  2. Ba=−b≠0\mathrm{a}=-\mathrm{b} \neq 0
  3. Ca=b≠0,h=k\mathrm{a}=\mathrm{b} \neq 0, \mathrm{ h}=\mathrm{k}
  4. Da=b≠0a=b \neq 0

Correct answer

B. a=−b≠0\mathrm{a}=-\mathrm{b} \neq 0

Step-by-step solution

dydx=ax+hby+k⇒(by+k)dy=(ax+h)dx\begin{aligned} & \frac{d y}{d x}=\frac{a x+h}{b y+k} \\ & \Rightarrow(b y+k) d y=(a x+h) d x\end{aligned}⇒by22+ky=ax22+hx⇒by2+2ky=ax2+2hx\begin{aligned} & \Rightarrow \frac{\mathrm{by}^2}{2}+\mathrm{ky}=\frac{\mathrm{ax}^2}{2}+\mathrm{hx} \\ & \Rightarrow \mathrm{by}^2+2 \mathrm{ky}=\mathrm{ax}^2+2 \mathrm{hx}\end{aligned}\Rightarrow \mathrm{ax}^2-\mathrm{by}^2+2 \mathrm{hx}-2 \mathrm{ky}=0ThisrepresentscircleonlywhenThis represents circle only whena=-b \neq 0$.
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