Question 1208377
<pre>
Find the equation of the circle passing through the points (1,2),(3,6),(5,4).

We start with: {{{matrix(1,3, (x  -  h)^2 + (y - k)^2, "=", r^2)}}} <==== Standard form of the equation of a CIRCLE: 

           <font size = 4><font color = blue><b><u>Point: (1, 2)</font></font></u></b>
          {{{matrix(1,3, (x  -  h)^2 + (y - k)^2, "=", r^2)}}} ---- Standard form of the equation of a CIRCLE
          {{{matrix(1,3, (1  -  h)^2 + (2 - k)^2, "=", r^2)}}} ---- Substituting (1, 2) for (x, y) 
1 - 2h + h<sup>2</sup> + 4 - 4k + k<sup>2</sup> = r<sup>2</sup>
    h<sup>2</sup> - 2h + k<sup>2</sup> - 4k + 5 = r<sup>2</sup> ----- eq (i)


            <font size = 4><font color = blue><b><u>Point: (3, 6)</font></font></u></b>
            {{{matrix(1,3, (x  -  h)^2 + (y - k)^2, "=", r^2)}}} ---- Standard form of the equation of a CIRCLE
            {{{matrix(1,3, (3  -  h)^2 + (6 - k)^2, "=", r^2)}}} ---- Substituting (3, 6) for (x, y) 
9 - 6h + h<sup>2</sup> + 36 - 12k + k<sup>2</sup> = r<sup>2</sup>
    h<sup>2</sup> - 6h + k<sup>2</sup> - 12k + 45 = r<sup>2</sup> ----- eq (ii)


            <font size = 4><font color = blue><b><u>Point: (5, 4)</font></font></u></b>
            {{{matrix(1,3, (x  -  h)^2 + (y - k)^2, "=", r^2)}}} ---- Standard form of the equation of a CIRCLE
            {{{matrix(1,3, (5  -  h)^2 + (4 - k)^2, "=", r^2)}}} ---- Substituting (5, 4) for (x, y) 
25 - 10h + h<sup>2</sup> + 16 - 8k + k<sup>2</sup> = r<sup>2</sup>
     h<sup>2</sup> - 10h + k<sup>2</sup> - 8k + 41 = r<sup>2</sup> ----- eq (iii)

       h<sup>2</sup> - 2h + k<sup>2</sup> - 4k + 5 = r<sup>2</sup> ----- eq (i)
     h<sup>2</sup> - 6h + k<sup>2</sup> - 12k + 45 = r<sup>2</sup> ----- eq (ii)
     h<sup>2</sup> - 10h + k<sup>2</sup> - 8k + 41 = r<sup>2</sup> ----- eq (iii)

              4h + 8k  -  40 = 0 ----- Subtracting eq (ii) from eq (i)
                     4h + 8k = 40
                   4(h + 2k) = 4(10)
                      h + 2k = 10 ---- eq (iv)

                 4h - 4k + 4 = 0 ----- Subtracting eq (iii) from eq (ii)
                     4h - 4k = - 4
                    4(h - k) = 4(- 1)
                       h - k = - 1 --- eq (v)

                      h + 2k = 10 ---- eq (iv)
                       h - k = - 1 --- eq (v)
                          3k = 11 ---- Subtracting eq (v) from eq (iv)
                          {{{matrix(1,3, k, "=",  11/3)}}}

                    h - {{{11/3}}} = - 1 ---- Substituting {{{11/3}}} for k in eq (v) 
                         {{{matrix(3,3, h, "=", - 1 + 11/3, h, "=", - 3/3 + 11/3, h, "=", 8/3)}}} 

        (x - h)<sup>2</sup> + (y - k)<sup>2</sup> = r<sup>2</sup>
         {{{matrix(1,3, (1  -  8/3)^2 + (2  -  11/3)^2, "=", r^2)}}} -- Substituting (1, 2) for (x, y), and ({{{matrix(1,3, 8/3, ",", 11/3)}}}) for (h, k)
         {{{matrix(4,3, (3/3  -  8/3)^2 + (6/3  -  11/3)^2, "=", r^2, (- 5/3)^2 + (- 5/3)^2, "=", r^2, 25/9 + 25/9, "=", r^2, 50/9, "=", r^2)}}}

            {{{matrix(1,3, (x  -  h)^2 + (y - k)^2, "=", r^2)}}} <==== Standard form of the equation of a CIRCLE
        {{{highlight(highlight_green(highlight(matrix(1,3, (x  -  8/3)^2 + (y - 11/3)^2, "=", 50/9))))}}} --- Substituting ({{{matrix(1,3, 8/3, ",", 11/3)}}}) for (h, k), and {{{50/9}}} for r<sup>2</sup></pre>