Question

An archeologist found the remains of an ancient spartans’s shield, which she then placed on a grid. If an arc of the shield passes through A(−6,0), B(2,4) and C(6,0), locate the center of the shield, and the standard equation defining its boundary.

Answer 1

Since we have 3 points, we will assume that the line's equation is this: $y=ax^2+bx+c$ .

We put the coordinates into this equation and we get:

$36a-6b+c=0$

$4a+2b+c=4$

$36a+6b+c=0$

$72a+2c=0$

$c=-36a$

$-6b=0$

$b=0$

$4a-36a=4$

$a=\frac{4}{-32}$

$a=$ $-\frac{1}{8}$

$c=-36\cdot(-\frac{1}{8})$

$c=\frac{9}{2}$

So the equation is $y=-\frac{1}{8}(x^2-36)$ .

Let's find a highest point of this parabola:

$x=\frac{-b}{2a}=0$

$y(0)=\frac{9}{2}$ .

Assuming that this equation that we got represents the upper part of the shield and the lower would go through points (-6,0), (2,-4), (6,0), having its equations as $y=\frac{1}{8}(x^2-36)$, we get that the center of the shield would be located right in middle between $(0,\frac{9}{2})$ and $(0,-\frac{9}{2})$, so it's (0,0).

Answer: $y=-\frac{1}{8}(x^2-36)$ - equation (for upper part), (0,0) - center