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Find all points (x,y) that are 5 units away from the point (2,7) and that lie on the line x + y = 13.

 Jun 13, 2024
 #1
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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! :)

 Jun 13, 2024

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