Circular Coordinate Geometry

Today I have a question involving some coordinate geometry, or analytic geometry. It also involves some circles. It's quite easy, but somewhat long. Here it comes.

What is the sum of the radii of all circles going through the points (1, 9) and (8, 8) and also tangent to the x-axis?

First, you should know that if a circle is tangent to the x-axis, its radius is the y-coordinate of its center. Let's call the center (x, y), so its radius is y.

Since it's a circle, the distance between the center and each of the points must be y. Let's use the distance formula to show that:

y = sqrt((x - 1)² + (y - 9)²)
y = sqrt((x - 8)² + (y - 8)²)

Square and open parentheses:
y² = x² - 2x + 1 + y² - 18y + 81
y² = x² - 16x + 64 + y² - 16y + 64

y^2 cancels on all equations (I also moved the y terms):
18y = x² - 2x + 82 /*8
16y = x² - 16x + 128 /*9

144y = 8x² - 16x + 656
144y = 9x² - 144x + 1152

9x² - 144x + 1152 = 8x² - 16x + 656
x² - 128x + 496 = 0
(x - 4)(x - 124) = 0
x = 4, 124

Now let's see what's y (the radius) for each x:
16y = 4² - 16*4 + 128 = 80
y = 5

When x = 124:
16y = 124² - 16*124 + 128 = 13520
y = 845

So the two circles have radii of 5 and 845, so the sum of the radii is 850.

Hope you liked it.
Nadav

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