Problem: Let PQ be a line segment of a fixed length L with it’s two ends P and Q sliding along the X axis and Y-axis respectively. Complete the rectangle OPRQ where O is the origin. Show that the locus of the foot of the perpendicular drawn from R on PQ is given by \(x^{\frac{2}{3}} + y^{\frac{2}{3}} + L^{\frac{2}{3}} \)

Discussion:

This beautiful problem discuss locus of an asteroid. An asteroid is a geometric figure which looks like this:

There are several locus definition of asteroid. A good reference book for a more detailed account is ‘Lines and Curves’ by Vasiliev. (This book is not available in print. We have an electronic copy, which we may give you for personal use).

Here we will work only on the problem (leaving the detailed discussion for class). This short video gives a visualization of the locus.

We will work with the following diagram:

Since length of PQ is L, if coordinate of Q = (0,h) then the coordinate of \(\displaystyle {P = (\sqrt {L^2 – h^2} ,0 )} \). Hence equation to line PQ is \(\displaystyle{\frac{x}{\sqrt {L^2 – h^2}} + \frac{y}{h} = 1} \).
Coordinate of R is \(\displaystyle{(\sqrt {L^2 – h^2}, h)} \).
We need equation to the line perpendicular from R to PQ. Clearly slope of that line will be negative reciprocal of the slope of PQ.

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