## Locus of PointFormed By Normal to Line From Origin Mobing Between Axes

A line of length
$L$
is free to move subject to the ends of the line being on the positive
$x$
and
$y$
axes.
A line drawn from the origin meets the line at a point M, such that the two lines are at right angles.

As the ends of the line move along the axes, what will be the curve described by the point M?

With the angle
$\alpha$
as above,
$x=\sqrt{x^2+y^2}cos \alpha , \: y= \sqrt{x^2 +y^2} sin \alpha$
.

\begin{aligned} L=L_1+L_2 &= \frac{x}{sin \alpha}+ \frac{y}{sin (90- \alpha )} \\ &= \frac{x}{sin \alpha}+ \frac{y}{cos \alpha} \\ &=\frac{x \sqrt{x^2+y^2}}{y}+ \frac{y \sqrt{x^2+y^2}}{y} \\ &= \frac{x^2 \sqrt{x^2+y^2}}{xy}+\frac{Y^2 \sqrt{x^2+y^2}}{xy} \\ &= \frac{ (\sqrt{x^2+y^2})^3}{xy} \end{aligned}

Then
$L^2x^2y^2=(x^2+y^2)^3$
.

Refresh