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Expert: - 12/3/2008


Question
Hi there, I have been stuck on a question for awhile
and I was wondering if you would be able to help?
I always get stuck on the equation of a line.

Peter has a summer job working for a landscaping company.  His boss has sent a plan to which includes a circular fountain. The rim of the fountain passes through grid points on the plan G(0, 14); D(5, 15) and B(18, 2) but the point to be used for the centre is not clear. Find the exact coordinates of the centre for Peter.


Answer
We must have equal distance between the center (x,y) and each of the points.  Since distance if the distances squared are equal if the distances are equal, I won't bother taking the squareroot for distance.  The three equations that must be the same are:

x² + (y-14)² = R²,
(x-5)² + (y-15)² = R²,  and
(x-18)² + (y-2)² = R².

Multiplying these out gives
x² + y² - 28y + 196 = R²,
x² - 10x + 25 + y² -30y + 225 = R², and
x² - 36x + 324 + y² - 4y + 4 = R².

Since all of the equations contain and x² + y², we will make a constant K = R² - x² - y² and then have
-28y + 196 = K,
-10x -30y + 250 = K, and
-36x - 4y + 328 = K.

Setting equation 1 equal to 2 and equation 1 equal to 3 gives
-28y + 196 = -10x - 30y + 250 and
-28y + 196 = -36x - 4y + 328.

Move the x, y terms to the right and the constants to the left, giving
10x + 2y = 54 and
36x - 24y = 132.

The first can be converted to 2y = 54 - 10x.  Multiply by -12 and substitue into the second, giving

36x - 648 + 120x = 132, which reduces to

156x = 780, or x = 5.  PUtting this back into either of the two equations where we had the two equations and two unknowns x and y wiil give you y = 2.

Looking at the graph again, this puts the center at (5,2).  We can see that D is 13 up, B is 13 to the right, and G is √(5²+12²)=√169=13.  This confirms the suspicion that (5,2) is at the center.

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