# Suppose a circle passes through (2,2) and (9,9) and touches the X - axis at P . If O is the origin then OP=

Arun
25750 Points
6 years ago
Write the general equation of a circle with center (h,k) and radius r:
(x - h)² + (y - k)² = r²

We know 3 points on this circle, the two points given and the tangent point on the x-axis:
(2,2)
(9,9)
(h,0)

Plug in each of these points:
(2 - h)² + (2 - k)² = r²
(9 - h)² + (9 - k)² = r²
(h - h)² + (0 - k)² = r²

We get k² = r² from the last equation, so let's replace that in the first two equations:
(2 - h)² + (2 - k)² = k²
(9 - h)² + (9 - k)² = k²

Expand both out:
4 - 4h + h² + 4 - 4k + k² = k²
81 - 18h + h² + 81 - 18k + k² = k²

Cancel k²:
4 - 4h + h² + 4 - 4k = 0
81 - 18h + h² + 81 - 18k = 0

Simplify:
h² - 4h + 8 - 4k = 0
h² - 18h + 162 - 18k = 0

We want to get rid of k, so multiply the first equation by 9 and the second equation by 2:
9h² - 36h + 72 - 36k = 0
2h² - 36h + 324 - 36k = 0

Subtract the second equation from the first:
7h² - 252 = 0

Divide both sides by 7:
h² - 36 = 0
h² = 36
h = ±√36

But we only care about the positive solution since the negative wouldn't make sense.