Exercise
Determine the envelope of the lines on which the coordinate axes detach a segment of constant length a.
Solution
We write the equation of the generic line of the assigned family. For this purpose, let's look at the figure:
Here we denote by φ the angle that the aforementioned straight line forms with the x axis. His equation is therefore:
The family equation follows:
where
By deriving with respect to φ
We have to solve the system of equations
From the second
Substituting in the first
Then the following parametric representation of the envelope:
It is a known curve: the asteroid (fig. 1).
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