[Math] Finding the value of $k$ for the equation of a circle

algebra-precalculuscircles

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.

Best Answer

A circle can meet an axis $0$, $1$ or $2$ times, so if it meets the two co-ordinate axes exactly three times in total then it must cross one axis at two points. The possibilities are:

  • it is tangent to the $y$-axis, and crosses the $x$-axis twice
  • it is tangent to the $x$-axis, and crosses the $y$-axis twice
  • it passes through the point $(0,0)$, crosses the $x$-axis at another point and crosses the $y$-axis at a third point

If you consider each of these three possibilities then you will have a complete solution.

Completing the squares imply the centre of the circle is at $(6,5)$ so these cases become:

  • possible, as having radius $6$ would satisfy the conditions
  • impossible, as having radius $5$ to be a tangent would then not cross the $y$-axis
  • possible, as having radius $\sqrt{6^2+5^2}$ would satisfy the conditions
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