Consider the picture below:
The circle with centre O touches the X- axis and the Y-Axis at B and A respectively.
Now, it can be easily shown that the radius is equal to the perpendicular distance of the centre to any of the axes. If the radius is $a$, then the equation for the circle in the first quadrant will be $(x-a)^2 + (y-a)^2 = a^2$
Now, the centre of such a circle will lie on the line:
- $y=x $ if the circle is in the first quadrant.
- $y=-x $ if the circle is in the second quadrant.
- $-y=-x $ if the circle is in the third quadrant.
- $-y=x $ if the circle is in the fourth quadrant.
If the equation of a circle is given, for example, take it to be $x^2 + y^2 + 6x - 6y +9=0$, then it can be easily said that the centre is $(-3,3) $ which lies on $y=-x $, which means that the circle touches the two co-ordinate axes.
Is it correct to infer this way?
N.B.: The equation of the circle I've used does not indicate this is a homework question. I just used it as an example.