A variable circle in the xy-plane is tangent to the x-axis and
meets the y-axis at the points P and Q. If the circle varies in such
a way that the length of the segment PQ is always 2, show that
the center A of the circle lies on a hyperbola, and find the equation of this
hyperbola.
For parameter 'a', the equation of the circle can be:
(x-a)^2 + (y - √(1+a^2))^2 = 1+a^2
The center is (a, √(1+a^2))
The radius is √(1+a^2)
The y intercepts are √(1+a^2) ± 1 which puts their separation at 2 units.
The parametric equation for the locus of points that can be the center of the circle is:
x = a
y = √(1+a^2)
Since x = a, substitute
y = √(1+x^2)
y^2 = 1+x^2
y^2 - x^2 = 1 with y>0 which is a hyperbola
See a graphic representation on Desmos:
https://www.desmos.com/calculator/2i1gfg6snv
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Posted by Larry
on 2024-02-06 12:36:26 |
Solution
