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Let's solve the problem step by step.

### Part (a): Write down the equation for the circle \( C \)

The circle touches the y-axis at the point \( (0, 9) \) and has a radius of 5.

- The center of the circle is \( (h, k) \), and since the circle touches the y-axis at \( (0, 9) \), the x-coordinate of the center must be 5 (the distance between the center and the y-axis is the radius). Therefore, the center is \( (5, 9) \).
- The equation of a circle is given by:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

Substituting the values \( h = 5 \), \( k = 9 \), and \( r = 5 \):

\[
(x - 5)^2 + (y - 9)^2 = 25
\]

Thus, the equation of the circle is:

\[
(x - 5)^2 + (y - 9)^2 = 25
\]

### Part (b): Find the length of \( PT \)

We are given that the point \( P(8, -7) \) is tangent to the circle at point \( T \). The length of the tangent from a point to a circle can be found using the following formula:

\[
\text{Length of the tangent} = \sqrt{d^2 - r^2}
\]

where \( d \) is the distance from point \( P(8, -7) \) to the center of the circle \( (5, 9) \), and \( r \) is the radius of the circle.

1. First, calculate the distance \( d \) between \( P(8, -7) \) and the center \( (5, 9) \) using the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting \( (x_1, y_1) = (5, 9) \) and \( (x_2, y_2) = (8, -7) \):

\[
d = \sqrt{(8 - 5)^2 + (-7 - 9)^2} = \sqrt{3^2 + (-16)^2} = \sqrt{9 + 256} = \sqrt{265}
\]

2. Now, use the tangent length formula:

\[
\text{Length of } PT = \sqrt{265 - 25} = \sqrt{240}
\]

\[
\text{Length of } PT = \sqrt{240} \approx 15.49
\]

Thus, the length of the tangent \( PT \) is approximately 15.49 units.

---

Would you like more details or clarifications on any part of this? Here are some follow-up questions that could deepen your understanding:

1. Why is the x-coordinate of the center of the circle equal to the radius in this case?
2. Can you derive the equation of the tangent line at point \( T \)?
3. What is the geometric significance of the distance formula in this context?
4. Can you explain why the tangent length formula includes the subtraction of \( r^2 \)?
5. What other methods can be used to verify the tangent length?

**Tip:** Always double-check your substitutions and calculations, especially when dealing with distance formulas or circle equations!

The circle C has radius 5 and touches the y-axis at the point (0, 9). A line through the point P(8, -7) is tangent to the circle C at point T. Write down the equation for the circle and find the length of PT.

Learn how to find the equation of a circle with radius 5, touching the y-axis at (0, 9), and how to calculate the length of the tangent from a point (8, -7) to the circle. This geometry problem covers the distance formula and properties of tangents.

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