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# ellipses

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In the coordinate plane, let  A = (-2,0) and B = (2,0) Let C be a variable point on the ellipse \frac{x^2}{4} + y^2 = 1
and let  H be the orthocenter of triangle ABC  As C varies on the ellipse, the point H  traces a curve. Find the area inside the curve.

May 22, 2022

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May 22, 2022
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The area is 12*pi.

May 22, 2022
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I have made a GeoGebra simulation of the problem, and it appears that the curve that H traces is an ellipse with major axis 4 units and minor axis 2 units.

GeoGebra simulation: https://www.geogebra.org/calculator/my8uamwy

Therefore, the area inside the curve is $$\pi \cdot 4 \cdot 2 = 8\pi\text{ square units}$$.

May 22, 2022