Official solution
EF = perpendicular bisector of chord AB
BG = perpendicular to y-axis
Here C = center of the circle
mid-point of chord AB, D = (- 1, 3)
slope of AB = \[{{4 - 2} \over { - 2 - 0}}\] = - 1
\[ \because \] EF \[ \bot \] AB
\[ \therefore \] Slope of EF = 1
Equation of EF, y - 3 = 1 (x + 1)
\[ \Rightarrow \] y = x + 4 . . . .(i)
Equation of BG
y = 2 . . . . (ii)
From equations (i) and (ii)
x = - 2, y = 2
since C be the point of intersection of EF and BG, therefore center, C = (- 2, 2)
Now coordinates of center C satiesfy the equation
2x - 3y + 10 = 0
Hence 2x - 3y + 10 = 0 is the equation of the diameter