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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 \(\mathcal{P}\) is a parabola with focus \((4,3)\) and directrix \(y=1\). The point \((8,6)\) is on \(\mathcal{P}\) because \( (8,6) \) is 5 units away from both the focus and the directrix.

If we write the equation whose graph is \(\mathcal{P}\) in the form \(y=ax^2 + bx + c\), then what is \(a+b+c\)?

Guest Oct 30, 2018
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The vertex  is  ( 4, 2)

 

We can start with this form

 

4p ( y - k)  = ( x - h)^2     where p  = 1  and ( h, k)  is the vertex

 

4p ( y - 2)  =  ( x - 4)^2   

 

4 ( y - 2)   = x^2 - 8x + 16

 

4y - 8  =  x^2 - 8x + 16

 

4y  =  x^2 - 8x + 24          divide both sides by 4

 

y = (1/4)x^2  - 2x + 6 

 

a = (1/4)      b  =  -2   and  c   =  6

 

And   their sum is     4 + 1/4   =    17  / 4  

 

 

cool cool cool

CPhill  Oct 30, 2018

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