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Hyperbolic center (Posted on 2024-02-06)

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.

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| Comment 1 of 3

Let A's y coordinate be y. The radius of the circle is also y.

Each of the two intersections of the circle with the y-axis is one unit away from that same y value, one above and one below, but the difference's square is always 1.

From the circle:

x^2 + 1 = y^2

y^2 - x^2 = 1, a hyperbola.

Edited on February 6, 2024, 12:34 pm

Posted by Charlie on 2024-02-06 12:33:07

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