• # question_answer Which one of the following differential equations represents the family of straight lines which are at unit distance from the origin? A) ${{\left( y-x\frac{dy}{dx} \right)}^{2}}=1-{{\left( \frac{dy}{dx} \right)}^{2}}$ B) ${{\left( y+x\frac{dy}{dx} \right)}^{2}}=1+{{\left( \frac{dy}{dx} \right)}^{2}}$ C) ${{\left( y-x\frac{dy}{dx} \right)}^{2}}=1+{{\left( \frac{dy}{dx} \right)}^{2}}$ D) ${{\left( y+x\frac{dy}{dx} \right)}^{2}}=1-{{\left( \frac{dy}{dx} \right)}^{2}}$

[c] $y=mx+c$ (Equation of straight lie) $\frac{dy}{dx}=m$ and $mx-y+c=0$ is at unit distance from origin. $\therefore \frac{\left| m(0)-(0)+c \right|}{\sqrt{{{m}^{2}}+{{(-1)}^{2}}}}=1\Rightarrow c=\sqrt{1+{{m}^{2}}}$ Now: ${{\left[ y-x\frac{dy}{dx} \right]}^{2}}={{[mx+c-xm]}^{2}}={{c}^{2}}=1+{{m}^{2}}$ also, ${{\left[ y+x\frac{dy}{dx} \right]}^{2}}={{[mx+c\,+mx]}^{2}}={{[2mx+\sqrt{1+{{m}^{2}}}]}^{2}}$ also, $1-{{\left( \frac{dy}{dx} \right)}^{2}}=1-{{m}^{2}}$ and $1+{{\left( \frac{dy}{dx} \right)}^{2}}=1+{{m}^{2}}$ $\Rightarrow {{\left[ y-x\frac{dy}{dx} \right]}^{2}}=1+{{\left( \frac{dy}{dx} \right)}^{2}}$