Problem 1: In the coordinate plane, any circle which passes through (-2,2) and (1,4) cannot also pass through?

Question

(x,2006)

What is the value of x?

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I feel like it has to do something with the radius, or that three points that are on a straight line can't simultaneously be in a circle, but I don't know what to do from there.

Answer 1

your 2nd comment is the one which is valid...so develop the line equation containing the 2 points , then set y = 2006 and solve for x

Answer 2

You definitely had the right idea:  3 points on a line cannot be on a circle (a line intersects a circle in at most 2 points)

write the equation of the the line passing through (-2,2) and (1,4)

m = 2/3

y = (2/3) x + b

use (1,4) to solve for b:

4 = (2/3)(1) + b

4 - 2/3 = 12/3 - 2/3 = 10/3 = b

the line through (-2,2) and (1,4) is y = (2/3) x + 10/3

now let y = 2006 and solve for x:

2006 = (2/3) x + 10/3

2006 - 10/3 = (2/3) x

(2006 - 10/3)(3/2) = x

3009 - 5 = x

x = 3004

therefore, (3004, 2006) cannot be on the circle passing through the first two points

x = 3004

Answer 3

Actually, the controlling geometric statement is that a circle can be drawn through any 3 points that are NOT on a line; in fact, this is a construction exercise.  So if you find the line that goes thru (-2,2) and (1,4), you can substitute y=2006 into it and find the x-value.