Expert: Sherry Wallin - 6/29/2011

Question
hi
i m not getting the answer of following question
"find the equation of a circle passing through (1,1) and(2,2)and whose radius is 1"

Answer
Samiksha~


There are many forms of equations for circles, the most common is (x-h)^2 + (y-k)^2 = r^2 where (h,k) is the center of the circle and r is the radius. You are given two different points that are on the circle and the radius. Use those points by substituting into the equation of the circle that I have given you:
(1-h)^2 + (1-k)^2 = 1^2  and  (2-h)^2 + (2-k)^2 = 1^2
You now have a system of two equations with two unknowns, so you can solve for (h,k) in the following way:
expand each equation ->
1-2h + h^2 + 1-2k + k^2 = 1 -> h^2-2h + k^2 - 2k = -1

and

4-4h+h^2 + 4-4k + k^2 = 1 -> h^2-4h + k^2 -4k = -7

now subtract the second from the first:

 h^2-2h + k^2 - 2k = -1
-h^2+4h - k^2 + 4k = 7
__________________________
         2h + 2k = 6  -> h + k = 3

now solve for h in terms of k:
h = 3-k
and substitute into either of the original equations
(3-k)^2 -2(3-k) + k^2 - 2k   = -1 -> 9 - 6k + k^2 - 6 + 2k + k^2 - 2k = -1 -> 2k^2 - 6k + 4 = 0 -> k^2 - 3k + 2 = 0 -> (k-1)(k-2) = 0
this implies that k = 1 or k = 2
If k = 1 then h = 2 and if k = 2 then h = 1  **You find this out when you substitute each value of k into either of the original equations

What does this all mean? It means that there are two equations for a circle that pass through the points (1,1) and (2,2) with a radius of 1.
To summarize: the general equation is either (x-1)^2 + (y-2)^2 = 1 or (x-2)^2 + (y-1)^2 = 1
Verify these two to be true by seeing if both points work in both equations.

Also you can graph the points (1,1) and (2,2). Since they are both on the circle and the radius is 1, start at (1,1) and move either up or down 1 unit which would put you at either (1,2) or (1,0). Likewise move to the right one unit and to the left one unit and you will be at either (2,1) or (2,0). Do the same thing starting from (2,2) and move up and down 1 unit and to the right and left 1 unit and you will find the points (2,3) or (2,1) and (3,2) and (1,2). Now look at where they intersect. They either have a center of (1,2) or (2,1) which is your (h,k).

Math Prof