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Geometrically speaking, a parabola is defined as the set of points that are the same distance from a given point and a given line. The point is called the focus of the parabola and the line is called the directrix of the parabola.

Suppose P is a parabola with focus (4,2) and directrix y=1. The point (8,6) is on P because (8,6) is 5 units away from both the focus and the directrix.

If we write the equation whose graph is P in the form y=ax^2+bx+c then what is a+b+c?

 Feb 16, 2020
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The vertex of the parabola will be found half-ways between the focus and the directrix.

The vertex will be the point (4, 1.5).

A formula for the parabola is:  y - k  =  a(x - h)2  where (h, k) are the coordinates of the vertex.

For this situation:  y - 1.5  =  a(x - 4)2.

Since the point  (8, 6)  is on the parabola:     6 - 1.5  =  a(8 - 4)2

                                                      --->      4.5  =  a(4)2     --->     4.5  =  16·a     --->     a  =  0.28125

 

The equation is:  y - 1.5  =  0.28125(x - 4)2.

Multiplying out:   y - 1.5  =  0.28125(x2 - 8x + 16)

                           y - 1.5  =  0.28125x2 - 2.25x + 4.5

                                   y  =  0.28125x2 - 2.25x + 6

 Feb 16, 2020

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