# show, using algebra, that the formula for a parabola is a special case of the formula for an ellipse

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The parametric form of an ellipse (centred on the origin) is given by

x = a sin t, y = b cos t  where t in [0,2pi] and a and b are constants.

The parametric form of a parabola (centred on the origin also) is given by

x = c u^2, y = 2cu  where u in [-infty,+infty] and c and d are constants.

Equating the x variables and y variables we get

1)  cu^2 = asint

2) 2cu = b cos t

Dividing 1) by 2) gives u = ((2a)/b) tan t

because  (sint)/(cost) = tan t

The range of t is now restricted to t in [-pi/2, pi/2] because the tangent is undefined outside this range. This gives u in [-infty,+infty] (and the values of a and b are irrelevant as they simply determine the "speed" with which we get to the ends of the parabola - which we never do!). Since the range of t is restricted to half it's usual range (for a full ellipse), this is why the parabola looks like half of an ellipse.

Written in parametric form, an ellipse and parabola are equivalent when we write

u = tan(t)   where t is the parameter of the ellipse and u the parameter of the parabola. The range of t is [-pi/2,pi/2] and the range of u is [-infty,+infty]

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