Question

If a circle passes through the point (0, 0),(a, 0),(0, b) then find the coordinates of its centre.

Answer 1

The general equation of the circle is x² + y² + 2gx + 2fy + c = 0
Now, it is passing through (0, 0)
∴ c = 0
Also, it is passing through (a, 0)
∴  a² + 2ag = 0
⇒ a(a + 2g) = 0
⇒a + 2g = 0

\[\Rightarrow g = - \frac{a}{2}\]

Again, it is passing through (0, b)
∴ b² + 2bf = 0
⇒ b(b + 2f) = 0
⇒b + 2f = 0

\[\Rightarrow f = - \frac{b}{2}\]

The coordinates of its centre are given by

\[\left( - g, - f \right) = \left( \frac{a}{2}, \frac{b}{2} \right)\]