Question

Find the equation of the circle passing through the points (1,-2) ,(5,4) and (10,5)​

Answer 1

First check whether, the given 3 coordinate pairs are in a single straight line. If so, no circle can be drawn passing through these. Else we can draw. Let the centre of that imaginary circle, A = (m, n). Let given 3 point are B, C, D in the given order.
As radi of a given circle is same,
Length AB = Length AC = Length AD

We have 2 simultaneous equations of m, n from above. Solve them and find m, n; coordinate pairs of the centre of the circle. From there onwards it's pretty simple maths to find the equation of that circle.

Alternatively you could use standard equation format of any circle which has 3 unknown variables to substitute aforementioned 3 points and solve the 3 simultaneous equations.

Answer 2

$The \: final \: circle \: equation \: by \: center</p><p>\\</p><p></p><p> \bold{(9,-3) and\:radius \sqrt{65}}</p><p>\\</p><p>is </p><p>\\</p><p>\bold{{(x-9)}^2+{(y+3)}^2={\sqrt{65}}^2=65}$

Hope it helps, if you've any doubts comment below