Of ellipses, hyperbolae and mugging

by ayushkhaitan3437

For as long as I can remember, I have had unnatural inertia in studying coordinate geometry. It seemed to be a pursuit of rote learning and regurgitating requisite formulae, which is something I detested. My refusal to “mug up” formulae cost me heavily in my engineering entrance exams, and I was rather proud of myself for having stuck to my ideals in spite of not getting into the college of my dreams.

However, now I realise what useful entities ellipses and hyperbolae are in reality. Hence, as a symbolic gesture, I will derive the formulae of both the ellipse and the hyperbola in the most simple settings- that of the centre being at the origin (0,0).

1. Ellipse- The sum of distances from two _foci_ is constant. Let the sum be “L“. As the centre is at the origin, and we are free to take the foci along the x-axis, the coordinates of the foci are (-c,0) and (c,0). We thus have the equation \sqrt{(x-c)^2 +y^2}+\sqrt{(x+c)^2+ y^2}=L. On simpifying this, we get \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, where a^2=\frac{L^2}{4} and b^2=\frac{(L^2-4c^2)}{4}.

2. In the case of a hyperbola, under similar conditions, we have the equation \sqrt{(x-c)^2 +y^2}-\sqrt{(x+c)^2+ y^2}=L. This under simplification gives \frac{x^2}{a^2}-\frac{y^2}{b^2}=1, where a^2=\frac{L^2}{4} and b^2=\frac{(4c^2-L^2)}{4}.  

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