sorry for the really basic question. i am looking at this old SAT question. what i am thinking is to plug in the coordinates of x and y into the equation of a circle for each coordinate and then solve a system of equations but that seems too tedious for an SAT question.is there any relationship between a chord and the origin? thanks for the help
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$\begingroup$One thing you could do is apply the Pythagorean theorem.
You know the horizontal distance between the origin and any of the two intersections (it is $(20-4)/2 = 8$).
Once you know that, you can build a right triangle between one of the intersections (say $(4,0)$), the origin and $(4,k)$ and solve for $k$ in the following formula
$$k^2 + 8^2 = 10^2$$
$\endgroup$ 2 $\begingroup$The equation of this circle is:$$(x-h)^2+(y-k)^2=100.$$ First Observe that $h,k>0$.Note that this circle passes through the point $(4,0)$ thus: $(4-h)^2+k^2=100$ also it passes through $(20,0)$ therefore $(20-h)^2+k^2=100$. Can you take it from here.
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