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Nitya Nigam There are four recognised types of conic sections: circles (probably the most familiar of the lot), ellipses, parabolas and hyperbolas. You’ve probably encountered the equation of a circle: (x-h)^2 + (y-k)^2 = r^2, where (h, k) is the circle’s centre, and r is its radius. In this article, we’ll derive the equations for ellipses and parabolas. Parabolas are defined as curves such that all points on the curve are equidistant from a point, called the focus, and a line, called the directrix. The diagram below displays this clearly: We can see that the vertical distance d from point (x, y) to line y = -p is y + p We can also see that the distance d from point (x, y) to the focus (0, p) is sqrt(x^2 + (y-p)^2) Equating these distances and rearranging gives the following: This is the standard equation for a parabola centred at (0, 0). As with the circle equation, we can shift parabolas around by incorporating the parameters (h, k) into the equation like so: And that is the parabola equation! Ellipses are defined as curves such that, for all points on the curve, the sum of the two distances to the two defining “focal points” is constant, and this sum is equal to the length of the major axis, which is the longer distance between the two “vertices” of the ellipse. This diagram provides a visual explanation: By Pythagoras’ equation, d_1 = sqrt[(x + c)^2 + y^2] and d_2 = sqrt[(c - x)^2 + y^2]. By applying this to the point (0, b) we obtain d_1 = d_2 = sqrt(c^2+ b^2). Substituting this into d_1 + d_2 = 2a gives: Using this information, we can use the general expressions for d_1 and d_2 to find an equation involving only x, y, a and b. This is the standard equation for an ellipse centred at (0, 0). As with the circle and parabola equations, we can shift ellipses around by incorporating the parameters (h, k) into the equation like so: And that is the ellipse equation!
Let us know in the comments if you found these derivations useful, and what you'd like to see next!
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