According to question, circle passes through (0, 0) and cut intercepts a and b at x and y-axis.
Intersection points of circle with x-axis will be (a, 0) and with y-axis will be (0, b).
Thus, circle will passed through three points (0, 0),(a, 0) and (0, b).
Let equation of circle,
(x – h)2 + (y – k)2 = r2 …(1)
Circle passes through (0, 0), (a, 0) and (0, b).
Thus, (0 – h)2 + (0 – k)2 = r2 …(i)
(a – h)2 + (0 – k)2 = r2 …(ii)
(0 – h)2 + (b – k)2 = r2 …(iii)
h2 + k2 = r2 …(2)
From eqn. (i),
From eqn. (ii),
a2 + h2 – 2 ah + k2 = r2
⇒ h2 + k2 – 2 ah + a2 = r2 …(3)
From eqn. (iii)
h2 + b2 + k2 – 2bk = r2
⇒ h2 + k2 – 2bk + b2 = r2 …(4)
From eq11. (2) and (3)
r2 – 2ah + a2 = r2
⇒ -2 ah + a2 – r2 – r2
⇒ a2 – 2h = 0
⇒ a(a -2h) = 0
Then a ≠ 0 and a – 2h = 0
⇒ h = a/2
Similarly, from eqn. (2) and (4)
r2 – 2bk + b2 = r2
⇒ – 2bk + b2 = r2 – r2
⇒ b2 – 2bk = 0
⇒ b(b – 2k) = 0
