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 .
1. Ellipse- The sum of distances from two _foci_ is constant. Let the sum be ““. 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 and . We thus have the equation . On simpifying this, we get , where and .
2. In the case of a hyperbola, under similar conditions, we have the equation . This under simplification gives , where and .