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.

locus of points formed by normal to line drawn between axes

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

locus of points formed by normal to line drawn between axes

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

locus of points formed by normal to line drawn between axes

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

Then  
\[L^2x^2y^2=(x^2+y^2)^3\]
.

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