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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 (4,3) 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 ?

May 27, 2020

#1
+1193
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The vertex is right between the focus and directrix, according to your definition.

Therefore, the vertex is at (4, 2).

That means the parabola passes through (4, 2).

It also passes through (8, 6), as said in the problem.

This means it also passes through (0, 6). This is because parabolas are symmetric, and this one is symmetric at x=4.

Now we have three points. We can find the parabolic equation.

Substituting (4, 2) into the equation we get $$2=16a+4b+c$$.

(8, 6) gets us $$6=64a+8b+c$$.

And finally, (0, 6) gets us $$6=c$$.

We have that c=6, so now the two equations are

$$16a+4b=-4 \\ 64a+8b=0$$

I'm sure you can solve this on your own; anyways, we get that $$a=-\frac{1}{4}$$ and that $$b=2$$.

Therefore, our equation is $$y=-\frac{1}{4}x+2x+6, \text{ so } a\cdot b\cdot c = -\frac{1}{4}\cdot 2\cdot 6=\fbox{-3}.$$

You are very welcome!

:P

May 27, 2020
#2
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Emmmm, the point is (4,3) not (4,2)

Guest May 27, 2020
#3
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I figured it out, it's y=1/4x-2x+6

May 27, 2020