From the figure
AD = b units and AE = a units.
D(0, b), E(a, 0) and A(0, 0) lies on the circle. C is the centre.
The general equation of a circle: (x - h)2 + (y - k)2 = r2 …(i),
where (h, k) is the centre and r is the radius.
Putting A(0, 0) in (i)
(0 - h)2 + (0 - k)2 = r2
h2 + k2 = r2 .....(ii)
Similarly putting D(0,b) in (i)
(0 - h)2 + (b - k)2 = r2
h2 + k2 + b2 - 2kb = r2
r2 + b2 - 2kb = r2
b2 - 2kb = 0
(b-2k)b = 0
Either b = 0 or k = \(\frac{b}{2}\)
Similarly putting E(a,0) in (i)
(a - h)2 + (0 - k)2 = r2
h2 + k2 + a2 - 2ha = r2
r2 + a2 - 2ha = r2
a2 = 2ha = 0
(a - 2h)a = 0
Either a 0 or h = \(\frac{a}{2}\)
Center = C\(\Big(\frac{a}{2},\frac{b}{2}\Big)\)
r2 = h2 + k2
r2 = \(\frac{a^2+b^2}{4}\)
Putting the value of r2, h and k in equation (i)
(x - h)2 + (y - k)2 = r2
\(\Big(x-\frac{a}{2}\Big)^2\) + \(\Big(y-\frac{b}{2}\Big)^2\) = \(\frac{a^2+b^2}{4}\)
x2 + y2 + \(\frac{a^2}{4}+\frac{b^2}{4}\) -xa - yb = \(\frac{a^2+b^2}{4}\)
x2 + y2 - xa - yb = 0
which is the required equation.
