math extra credit?

Question

Consider the circles that have radii 4(squareroot of 5) and are tangent to the line x - 2y = 20 at the point (6, -7). Find the sum of the x coordinates of the centers of the circles.

Answer 1

Well, let's work this out... If a circle is tangent to the line at (6, -7) then it is 4 * sqrt(5) distance from that point. A line from the center to the point (6, -7) will be at a right angle to the line. Therefore, there are two such circles, one above the line and one below.

Now let's consider this a second... There will be two x coordinates, one to the left of (6, -7) and one to the right of (6, -7). Because you are using circles tangent to the line, the centers will form a line at a right angle to x - 2y = 20.

Now, if you go the same distance of 4*sqrt(5) up that line and down that line, you will go the same distance one way or the other way along the x-axis. Let's call that distance A.

The two x coordinates will be (6-A) and (6+A). If you add these two coordinates, you get 6-A+6+A = 12

So, the sum is 12.