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# FP1 Parabola doubt watch

1. I've previously learnt that this is the general equation of a parabola :
1/2(b-k) * (x-a)^2 = y-((b+k)/2)

But my current book says that it is represented by
y^2=4ax

I was wondering if someone could confirm/show that they are the same thing...
2. Well any graph of the form or is a parabola, so both your equations give parabolae.

However we can consider parabolae that go through the origin. If the directrix is the line x=-a and the focus is the point (a,0), then at any point on the graph, the horizontal distance from the directrix (i.e. ) must be equal to the distance from the focus (i.e. ). That is:

Squaring both sides and rearranging then gives .

Then we can drop the requirement that the parabola go through the origin by translating, meaning that a general parabola is given by , which can be rearranged into the form . We can then switch x and y if we need to.

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