Question

Find the equation of the circle passing through the points:

(5, 7), (8, 1) and (1, 3)

Answer 1

Let the required circle be

\[x^2 + y^2 + 2gx + 2fy + c = 0\]  ...(1)

It passes through (5, 7), (8, 1) and (1, 3).

Substituting the coordinates of these points in equation (1):

\[74 + 10g + 14f + c = 0\] ...(2)

\[65 + 16g + 2f + c = 0\] ...(3)

\[10 + 2g + 6f + c = 0\]...(4)

Simplifying (2), (3) and (4):

\[g = \frac{- 29}{6}, f = \frac{- 19}{6}, c = \frac{56}{3}\]

Equation of the required circle:

\[x^2 + y^2 - \frac{29x}{3} - \frac{19y}{3} + \frac{56}{3} = 0\]

\[\Rightarrow\] \[3\left( x^2 + y^2 \right) - 29x - 19y + 56 = 0\]