+228 Find all points (x,y) that are 5 units away from the point (2,7) and that lie on the line x + y = 13.
+1826 All the points 5 units away from (2, 7) form a nice little circle.
The formula for the circle is \((x-2)^2+(x-7)^2=25\).
We now have a system of equations with
\((x-2)^2+(x-7)^2=25\\ x+y=13\)
Let's focus on the first equation. Factoring it, we get
\(2x^2-18x+53=25\\ 2x^2-18x+28=0\\ x^2-9x+14=0\\ (x-7)(x-2)=0\\ x=7,2\)
Plugging these two values into the second equation, we get
\(y=13-2=11\\ y=13-7=6\)
So our two solutions are
\(\begin{pmatrix}x=2,\:&y=11\\ x=7,\:&y=6\end{pmatrix}\)
So (2,11) and (7,6) are our answers!
Thanks! :)
