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A point P moving on an ellipse of eccentricity e is connected to the two foci S1 and S2. The incenter of triangle PS1S2 is located on
- the major axis of the ellipse
- a circle of radius e
- an ellipse with eccentricity √((3+e²)/4)
- none of these
Correct answer: none of these
Solution
The incenter I=(c cos t, c*sqrt(a^2-c^2) sin t/(a+c)) traces an ellipse with semi-axes c and c*sqrt(a^2-c^2)/(a+c); its eccentricity satisfies e_I^2 = 2e/(1+e), which does not match sqrt((3+e^2)/4). So none of the given options is correct.
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