Question 923980

I just want to check if my answer is correct.

The problem is: Find the center of the circle passing through (9,-2),(2,5),(-3,6).
My answer is: (-3,-7)
<pre>
That's correct!!

Plug each point into the center-radius equation, since each radius will have the same center point
(h, k), and radius, r
<font face = "Tohoma" size = 4 color = "indigo"> <b>
(9, - 2)</font face = "Tohoma" size = 4 color = "indigo"> </b>
{{{(x - h)^2  + (y - k)^2 = r^2}}} 
{{{(9 - h)^2  + (- 2 - k)^2 = r^2}}}
{{{81 - 18h + h^2 + 4 + 4k + k^2 = r^2}}}
{{{h^2 - 18h + k2 + 4k - r^2 = - 85}}} ------- eq (i)
<font face = "Tohoma" size = 4 color = "indigo"> <b>
(2, 5)</font face = "Tohoma" size = 4 color = "indigo"> </b>
{{{(x - h)^2  + (y - k)^2 = r^2 }}}
{{{(2 - h)^2  + (5 - k)^2  =  r^2 }}}
{{{4 - 4h + h^2 + 25 - 10k + k^2 = r^2}}}
{{{h^2 - 4h + k^2 - 10k - r^2 + 4 + 25 = 0}}}
{{{h^2 - 4h + k^2 - 10k - r^2 = - 29}}} ------- eq (ii)
<font face = "Tohoma" size = 4 color = "indigo"> <b>
(- 3, 6)</font face = "Tohoma" size = 4 color = "indigo"> </b>
{{{(x - h)^2  + (y - k)^2 = r^2}}}
{{{(- 3 - h)^2  + (6 - k)^2  =  r^2 }}}
{{{9 + 6h + h^2 + 36 - 12k + k^2 = r^2}}}
{{{h^2 + 6h + k^2 - 12k - r^2 + 9 + 36 = 0}}}
{{{h^2 + 6h + k^2 - 12k - r^2 = - 45}}} ------- eq (iii)

{{{h^2 - 18h + k2 + 4k - r^2 = - 85}}} ------- eq (i)
{{{h^2 - 4h + k^2 - 10k - r^2 = - 29}}} ------ eq (ii)
– 14h + 14k = - 56 ------ Subtracting eq (ii) from eq (i)
– 14(h - k) = - 14(4) 
h - k = 4 ----------- eq (iv)

{{{h^2 - 4h + k^2 - 10k - r^2 = - 29}}} ------ eq (ii)
{{{h^2 + 6h + k^2 - 12k - r^2 = - 45}}} ------ eq (iii)
– 10h + 2k = 16 ------ Subtracting eq (iii) from eq (ii) 
- 2(5h - k) = - 2(- 8)
5h - k = - 8 --------- eq (v)

   h - k =   4 -------- eq (iv)
  5h - k = - 8 -------- eq (v) 
- 4h     =  12 -------- Subtracting eq (v) from eq (iv)
  {{{h = 12/(- 4)}}}, or – 3 

- 3 - k = 4 ----- Substituting - 3 for h in eq (iv) 
    - k = 4 + 3
    – k = 7
      {{{k = 7/(- 1)}}}, or - 7
With h = - 3, and k = - 7, center coordinate point (h, k) is: {{{highlight_green(highlight_green(system (h = - 3, k = - 7)))}}}