SOLUTION: Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose centre is on the line 4x+y=16.

Algebra ->  Circles -> SOLUTION: Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose centre is on the line 4x+y=16.      Log On


   

Question 738620: Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose centre is on the line 4x+y=16.
Answer by fcabanski(1391) About Me  (Show Source):
You can put this solution on YOUR website!
The standard equation for a line is %28x-h%29%5E2+%2B+%28y-k%29%5E2+=+r%5E2 where (h.k) is the center and r is the radius.


(4,1) lies on the circle, so %284-h%29%5E2+%2B+%281-k%29%5E2+=+r%5E2 Call that eq: 1.


(6,5) lies on the circle, so %286-h%29%5E2+%2B+%285-k%29%5E2+=+r%5E2 Call that eq: 2


(h,k) is the center of the circle, and therefore must lie on the line that passes through the center. 4h + k = 16 and thus k = 16-4h eq: 3


Set 1 = 2 since both = r squared. %284-h%29%5E2+%2B+%281-k%29%5E2+=%286-h%29%5E2+%2B+%285-k%29%5E2


Expand the squared binomials: 16+-+8h+%2B+h%5E2+%2B+1+-2k+%2B+k%5E2+=+36+-+12h+%2B+h%5E2+%2B+25+-10k+%2B+k%5E2.


Subtract h squared and k squared from both sides, and combine like terms.


17 - 8h - 2k = 61 -12h - 10k


4h = 44 - 8k so h = 11-2k Substitute that into eq:3 and solve it for k


k = 16 - 4(11-2k) = 16 - 44 +8k = 28+8k


-7k =28


k = -4


h = 11-2k = 11-2(-4) = 11+8 = 19.


The center of the circle is (19,-4) and thus the equation is:


%28x-19%29%5E2+%2B+%28y-%28-4%29%29%5E2+=+r%5E2


Use one of the given points to find r. Use (4,1).


%284-19%29%5E2+%2B+%281%2B4%29%5E2+=+r%5E2+=+%28-15%29%5E2+%2B5%5E2+=+225+%2B+25+=+250
The equation is %28x-19%29%5E2+%2B+%28y-%28-4%29%29%5E2+=+250

Hope the solution helped. Sometimes you need more than a solution. Contact fcabanski@hotmail.com for online, private tutoring, or personalized problem solving (quick for groups of problems.)