Find the equation of a circle
which touches both the axes and passes through the point (2, 1).
Let (h, k) be the centre of a circle with radius a.
Thus, its equation will be \[\left( x - h \right)^2 + \left( y - k \right)^2 = a^2\]
Let the required equation of the circle be
It is given that the circle touches both the axes.
Thus, the required equation will be
Also, the circle passes through the point (2, 1).
∴ \[4 + 1 - 4a - 2a + a^2 = 0\]
\[\Rightarrow a^2 - 6a + 5 = 0\]
\[ \Rightarrow a^2 - 5a - a + 5 = 0\]
\[ \Rightarrow a = 1, 5\]
Hence, the required equation is \[x^2 + y^2 - 2x - 2y + 1 = 0\] or