Question

Write the coordinates of the centre of the circle passing through (0, 0), (4, 0) and (0, −6).

Answer 1

We need to find the coordinates of the centre of the circle passing through (0, 0), (4, 0) and (0, −6).

Let the equation of the circle be \[\left( x - h \right)^2 + \left( y - k \right)^2 = a^2\] .

Putting x = y = 0:

\[h^2 + k^2 = a^2\]    ...(1)

Putting x = 4, y = 0 in the equation of the circle:

\[\left( 4 - h \right)^2 + \left( 0 - k \right)^2 = a^2 \]

\[ \Rightarrow 16 + h^2 - 8h + k^2 = a^2 \]

\[ \Rightarrow 16 - 8h + a^2 = a^2 \left( \text
{ From } (1) \right)\]

\[ \Rightarrow h = 2\]

Putting x = 0, y = −6 in the equation of the circle:

\[\left( 0 - h \right)^2 + \left( - 6 - k \right)^2 = a^2 \]

\[ \Rightarrow 36 + h^2 + 12k + k^2 = a^2 \]

\[ \Rightarrow 36 + 12k + a^2 = a^2 \left( \text { From } (1) \right)\]

\[ \Rightarrow k = - 3\]

Hence, the centre of the circle is \[\left( 2, - 3 \right)\] .