[alogo] Polar property

Given the ellipse (e) with center at H, axes a=HE, b=HI (a>b) and equation x/a + y/b =1. The auxiliary circle (c) of the ellipse is the one with diameter DE. For each point L on the ellipse the vertical line LO intersects the auxiliary circle at a point O, defining the [eccentric angle] phi = angle(EHO) of point L. For the coordinates (x,y) of L holds x = a*cos(phi) and y = b*sin(phi) (see Auxiliary.html ). Consider a point K exterior to the ellipse, the line HK cutting (e) at Q and define the ratio r = HQ/HK. Then the following is true. The polar line (k) of K intersects (e) at two points L, M such that the corresponding radii define angles angle(OHP) = angle(PHN) = phi, with cos(phi) = r = HQ/HK.

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The proof follows easily by expressing the points Q, L, M in terms of their eccentric angles Q = (a*cos(u), b*sin(u)) etc. , taking into account that the equation of the polar is

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See Also

Auxiliary.html
Ellipse.html

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