Rights In An Ellipse

Geometry Level pending

Suppose that f 1 f_1 and f 2 f_2 are the foci of the ellipse. What is the length of A f 2 ‾ \overline{Af_2} , in terms of the semi-major axis a a and the semi-minor axis b b ?

b 2 a \frac{b^2}{a} b a \frac{b}{\sqrt{a}} b a \frac{b}{a} b 2 a 2 \frac{b^2}{a^2}

Andrei Li by Andrei Li , 16, Canada

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1 solution

Let

a = semi-major axis

b = semi-minor axis

2s = length f 1 f 2 ‾ \overline{f_1 f_2}

h = length A f 1 ‾ \overline{Af_1}

∴ a 2 − s 2 = b 2 \therefore a^2 - s^2 = b^2

and

( 2 a − h ) 2 − ( 2 s ) 2 = h 2 (2a-h)^2 - (2s)^2 = h^2

4 a 2 − 4 a h + h 2 − 4 s 2 = h 2 4a^2 - 4ah + h^2 - 4s^2 = h^2

b 2 = a h b^2 = ah

∴ h = b 2 a \therefore h = \frac{b^2}{a}

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