A normal is drawn at a point $P(x, y)$ of a curve meets the $x$ - axis at $Q$. If $P Q$ is of consta

Question

A normal is drawn at a point $P(x, y)$ of a curve meets the $x$ - axis at $Q$. If $P Q$ is of constant length $K$, then the differential equation of the curve is

Professor PC Thomas Classes · Answer

$\frac{d y}{d x}$ denotes the slope of the tangent at $P=M$ (say)\ $	herefore \quad$ Equation of the normal at $P$ is\ $Y-y=-\frac{1}{M}(X-x)$\ This meets the $x$-axis at $Y=0$\ $\Rightarrow \quad X=x+M y$\ $	herefore \quad P(x, y)$ and $Q(x+M y, 0)$\ As $P Q=K, P Q^{2}=K^{2}$\ $	herefore \quad(M y)^{2}+y^{2}=K^{2}$\ $\Rightarrow \quad M y= \pm \sqrt{K^{2}-y^{2}}$\ or $\quad y \frac{d y}{d x}= \pm \sqrt{K^{2}-y^{2}}$.\