The diagram below shows a circle with centre P and equation 𝑥2+𝑦2−6𝑥−4𝑦=12. The

circle cuts the 𝑦−𝑎𝑥𝑖𝑠 at R and S. Tangent TS cuts 𝑥−𝑎𝑥𝑖𝑠 at T     Show that the coordinates of S are 𝑆(0,−2)

To show that the coordinates of point \( S \) are \( S(0, -2) \) using the given equation of the circle, let's follow these steps:

### 1. **Equation of the Circle**

The equation of the circle is given as:
\[
x^2 + y^2 - 6x - 4y = 12
\]
First, we need to rewrite this equation in the standard form of a circle's equation by completing the square.

### 2. **Complete the Square**

Group the \( x \)-terms and the \( y \)-terms:
\[
(x^2 - 6x) + (y^2 - 4y) = 12
\]

To complete the square:

- For the \( x \)-terms: \( x^2 - 6x \), take half the coefficient of \( x \) (which is \(-6\)), square it, and add it inside the bracket. \((-6/2)^2 = 9\).
- For the \( y \)-terms: \( y^2 - 4y \), take half the coefficient of \( y \) (which is \(-4\)), square it, and add it inside the bracket. \((-4/2)^2 = 4\).

Add these squares to both sides of the equation:

\[
(x^2 - 6x + 9) + (y^2 - 4y + 4) = 12 + 9 + 4
\]

This simplifies to:

\[
(x - 3)^2 + (y - 2)^2 = 25
\]

So, the equation of the circle is:

\[
(x - 3)^2 + (y - 2)^2 = 5^2
\]

This represents a circle with:

- **Center \( P(3, 2) \)**
- **Radius \( 5 \)**

### 3. **Finding Points \( R \) and \( S \) on the \( y \)-Axis**

The circle cuts the \( y \)-axis at points \( R \) and \( S \). On the \( y \)-axis, the \( x \)-coordinate is 0, so substitute \( x = 0 \) into the circle's equation to find \( y \):

\[
(0 - 3)^2 + (y - 2)^2 = 25
\]
\[
9 + (y - 2)^2 = 25
\]
\[
(y - 2)^2 = 16
\]
\[
y - 2 = \pm 4
\]

So:

\[
y = 2 + 4 = 6 \quad \text{or} \quad y = 2 - 4 = -2
\]

The coordinates of the points where the circle cuts the \( y \)-axis are \( R(0, 6) \) and \( S(0, -2) \).

### 4. **Conclusion**

Thus, we have shown that the coordinates of \( S \), where the circle intersects the \( y \)-axis, are \( S(0, -2) \).

Would you like more details or have any questions? Here are some related questions:

1. How do you find the equation of a circle given its center and radius?
2. What is the process for completing the square to rewrite a quadratic equation?
3. How can you determine the points of intersection between a circle and an axis?
4. What is the geometric significance of the radius and center of a circle?
5. How do you find the equation of a tangent to a circle from a given point?

**Tip:** Completing the square is a useful technique for converting the general form of a circle's equation to its standard form, making it easier to find the center and radius.

Learn how to determine the coordinates of point S on the y-axis of a circle using circle geometry and completing the square. This problem involves analyzing a circle with a given equation and finding specific intersection points. Suitable for high school students studying coordinate geometry.

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