Question

Find the orthogonal trajectories of the family of straight lines- $y=mx+1$.

• A. $x^2+y^2=c$
• B. $x^2+y^2-2y=c$
• C. $x^2+y^2-4y=c$
• D. $x^2+2y^2-2y=c$

Answer 1

The correct option is B $x^2+y^2-2y=c$

Given equations of the family of straight lines is $y=mx+1$

Here $m$ is the arbitrary constant or parameter.

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

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

Substituting the value of $m$ in the equation gives.,

$y=x\dot{y}+1$

Now,to find the equation of the orthogonal trajectory wwe need to replace $\frac{dy}{dx}$ by $-\frac{dx}{dy}$ which gives.,

$(y-1)dy=-xdx$

Integrating both sides of the equation gives.,

$\frac{y^2}{2}-y=-\frac{x^2}{2}+c$

Rearranging the terms gives.,

$x^2+y^2-2y=c$, where $c$ is a constant.