I have been told that the circle with equation $x^2 + y^2 - 12x -10y + k=0$ meets the co-ordinate axes exactly three times, and I have to find the value of $k$.
Now, I first found the centre of the circle, with the information given, to be $(6,5)$, and substituing this into the equation, we obtain $k=61$. Yet, $k$ cannot equal $61$ since that would imply the radius of the circle is zero, a contradiction to the fact that the equation is a circle. From this, I concluded that $k=0$ (the answer in the marking instructions), yet the marking instructions does not state my solution (although, I do know it is not correct).
In the marking instructions, there are two solutions, $k=25$ and $k=0$, and they are found, respectively, by assuming that the circle is tangent to the y-axis and from this calculating the radius of the circle (which would then provide the value of $k$), or that the circle touches the origin and from this calculating the radius of the circle. Now, I don't know if their solutions are correct or not, because they don't exactly show that their obtained value of $k$ satisfies the condition on the circle (that it meets the co-ordinate axes exactly three times).
My questions are whether these solutions are the only solutions and and whether it's possible to show that they are indeed the only solutions.