# Is this a good reason to prove that a circle touches both the co-ordinate axes?

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:

1. $y=x$ if the circle is in the first quadrant.
2. $y=-x$ if the circle is in the second quadrant.
3. $-y=-x$ if the circle is in the third quadrant.
4. $-y=x$ 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.

• If you put say $x=0$ into the equation $x^2+y^2+6x-6y+9=0$ it reduces to $(y-3)^2=0$. This has a double root, so $x=0$ is a tangent... – Lord Shark the Unknown Sep 22 '17 at 5:11
• OK, that is a good way.... – Wrichik Basu Sep 22 '17 at 5:12
• Note that $y=x$ and $-y=-x$ are the same, you should better talk about $y=x,\ x\ge 0$ or $y=x,\ x\le 0$ to locate quadrants. Anyway, in your inference, saying that the center is $(-3,3)$ is not enough to conclude, you need also to know $r=3$, if $r<3$ it won't touch axes, and if $r>3$ it will cross the axes. – zwim Sep 22 '17 at 5:36
• @zwim if $r > 3$ or $r < 3$, won't the equation change? – Wrichik Basu Sep 22 '17 at 5:39
• Yes, equation will change, but you still need to calculate $r$ to be certain that the circle is tangent to axes. Or use the trick given by LSU. – zwim Sep 22 '17 at 5:42

Your inference is fully incorrect !

If following hold true :

1. Circle touches both co-ordinate axes.

Then following will hold true :

1. Radius of circle will be equal to perpendicular distance of center of circle from either of the axes.

2. Center of circle will lie in either of the following lines : y = x | y = -x.

Implication doesn't hold the other way round !

That is, following result is incorrect.

If following hold true :

1. Center of circle will lie in either of the following lines : y = x | y = -x.

Then following will hold true :

1. Circle touches both co-ordinate axes.

You can consider following contradictory example :

x2 + y2 + 6*x − 6*y = 0, through center lies on y = -x, but this circle intersects with X & Y axes.