Question

Which one of the following differential equations represents the family of straight lines which are at unit distance from the origin?

• A. ${(y-x\frac{dy}{dx})}^2=1-{\left(\frac{dy}{dx}\right)}^2$
• B. ${(y+x\frac{dy}{dx})}^2=1+{\left(\frac{dy}{dx}\right)}^2$
• C. ${(y-x\frac{dy}{dx})}^2=1+{\left(\frac{dy}{dx}\right)}^2$
• D. ${(y+x\frac{dy}{dx})}^2=1-{\left(\frac{dy}{dx}\right)}^2$

Answer 1

The correct option is A ${(y-x\frac{dy}{dx})}^2=1-{\left(\frac{dy}{dx}\right)}^2$

Consider the equations of the family of straight lines be $y=mx+c$,

Given that they are at a unit distance from origin,

$⟹c=\sqrt{1+m^2}$,

Substituting the value of $c$ gives,

$y=mx+\sqrt{1+m^2}$,Which has only one arbitrary constant i.e.,$m$,

Now,differentiating the equation with respect to $x$ gives,

$\frac{dy}{dx}=m$,

$\therefore y=\frac{dy}{dx}x+\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}$

$\therefore {(y-x\frac{dy}{dx})}^2=1+{\left(\frac{dy}{dx}\right)}^2$ is the differential equation.