Question

A normal is drawn at a point $P(x,y)$ of a curve. It meet the $x-$axis at $Q.$ If $PQ$ is of constant length $k,$ then the differential equation describing such curves is:

• A. $\frac{ydy}{dx}=\pm \sqrt{k^2-x^2}$
• B. $\frac{ydy}{dx}=\pm \sqrt{k^2-y^2}$
• C. $\frac{xdy}{dx}=\pm \sqrt{k^2-x^2}$
• D. $\frac{xdy}{dx}=\pm \sqrt{k^2-y^2}$

Answer 1

The correct option is B $\frac{ydy}{dx}=\pm \sqrt{k^2-y^2}$

$PQ$ is equal to length of normal of the curve $y=f\left(x\right)$ which is given by
$y\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}$
According to question length of $PQ=k$
$\Rightarrow {\left(y\frac{dy}{dx}\right)}^2+y^2=k^2$
$\Rightarrow y\frac{dy}{dx}=\pm \sqrt{k^2-y^2}$
which is the required differential equation of given curves.