To solve this problem, let's start by understanding the given information and breaking down the steps:

1. Circle's Equation

The given equation of the circle is:

$x^2 + y^2 - 6x - 4y = 12$

To find the center and radius of the circle, we need to rewrite this equation in standard form by completing the square for both $x$ and $y$.

Completing the Square:

1. Group $x$ and $y$ terms:

   $(x^2 - 6x) + (y^2 - 4y) = 12$

2. Complete the square for $x$:

   • For $x^2 - 6x$, half of $-6$ is $-3$, and $(-3)^2 = 9$.
   • Add and subtract 9: $(x^2 - 6x + 9) - 9$.

3. Complete the square for $y$:

   • For $y^2 - 4y$, half of $-4$ is $-2$, and $(-2)^2 = 4$.
   • Add and subtract 4: $(y^2 - 4y + 4) - 4$.

4. Rewrite in completed square form:

   $(x - 3)^2 - 9 + (y - 2)^2 - 4 = 12$

   Simplifying:

   $(x - 3)^2 + (y - 2)^2 = 25$

Circle's Center and Radius:

• The center of the circle, $P$, is at $(3, 2)$.
• The radius, $r$, is $\sqrt{25} = 5$.

2. Points R and S on the y-axis:

To find where the circle intersects the y-axis, set $x = 0$ in the circle's equation:

$(0 - 3)^2 + (y - 2)^2 = 25$

$9 + (y - 2)^2 = 25$

$(y - 2)^2 = 16$

$y - 2 = \pm 4$

$y = 6 \text{ or } y = -2$

Thus, the points where the circle intersects the y-axis are $R = (0, 6)$ and $S = (0, -2)$.

3. Tangent Line TS:

The tangent line $TS$ passes through $S$ and is perpendicular to the radius at point $S$.

• At $S = (0, -2)$, the slope of the radius $PS$ is:

  $\text{slope of } PS = \frac{-2 - 2}{0 - 3} = \frac{-4}{-3} = \frac{4}{3}$

• The slope of the tangent $TS$ is the negative reciprocal:

  $\text{slope of } TS = -\frac{3}{4}$

Using the point-slope form for the tangent line $TS$ passing through $S = (0, -2)$:

$y + 2 = -\frac{3}{4}x$

Simplifying:

$y = -\frac{3}{4}x - 2$

4. Tangent TS Intersects x-axis at T:

To find where the tangent $TS$ intersects the x-axis ($y = 0$):

$0 = -\frac{3}{4}x - 2$

$\frac{3}{4}x = -2$

$x = -\frac{8}{3}$

So, point $T$ is $\left(-\frac{8}{3}, 0\right)$.

---

Would you like to see more details or have any questions? Here are some related questions to explore further:

1. How can you verify that $TS$ is perpendicular to the radius $PS$ at point $S$?
2. What is the equation of the line passing through points $R$ and $S$?
3. How would the problem change if the circle's center or radius were different?
4. What would be the coordinates of $T$ if the tangent line was drawn at $R$ instead of $S$?
5. How can you find the length of $TS$ using coordinate geometry?

Tip: Always double-check calculations when completing the square to ensure the circle's equation is accurate.