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?

Hi121529 Feb 16, 2020

#1**0 **

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(x^{2} - 8x + 16)

y - 1.5 = 0.28125x^{2} - 2.25x + 4.5

y = 0.28125x^{2} - 2.25x + 6

geno3141 Feb 16, 2020