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For question 14 on the sheet:

### Problem Statement (Question 14)
The centers of circles with equations:
1. \((x + 8)^2 + (y - 8)^2 = 117\)
2. \((x + 1)^2 + (y - 3)^2 = 106\)

are denoted as points \(P\) and \(Q\) respectively.

#### Part a:
Find the coordinates of \(P\) and \(Q\).

#### Part b:
Calculate the length of \(PQ\).

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### Solution

#### Part (a) - Finding Coordinates of \(P\) and \(Q\)

1. **Circle 1**: \((x + 8)^2 + (y - 8)^2 = 117\)
   - The general form of a circle equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
   - Comparing \((x + 8)^2 + (y - 8)^2 = 117\) to this form, we find:
     - Center \(P\) is at \((-8, 8)\).

2. **Circle 2**: \((x + 1)^2 + (y - 3)^2 = 106\)
   - Using the same approach:
     - Center \(Q\) is at \((-1, 3)\).

So, the coordinates are:
- \(P = (-8, 8)\)
- \(Q = (-1, 3)\)

#### Part (b) - Calculating the Distance \(PQ\)

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting the coordinates of \(P\) and \(Q\):
- \(P = (-8, 8)\)
- \(Q = (-1, 3)\)

\[
PQ = \sqrt{((-1) - (-8))^2 + (3 - 8)^2}
\]
\[
= \sqrt{(7)^2 + (-5)^2}
\]
\[
= \sqrt{49 + 25}
\]
\[
= \sqrt{74}
\]
\[
\approx 8.6
\]

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### Final Answer

- **Coordinates of \(P\)**: \((-8, 8)\)
- **Coordinates of \(Q\)**: \((-1, 3)\)
- **Length of \(PQ\)**: \(\sqrt{74} \approx 8.6\)

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Would you like a more detailed explanation or assistance with another question?

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### Related Questions

1. How do we determine the radius of each circle from the given equations?
2. How do we use the general form of the circle equation to find other circle properties?
3. Can we find the midpoint between \(P\) and \(Q\) and interpret its meaning in relation to the circles?
4. How would we set up the equation for a line passing through \(P\) and \(Q\)?
5. How can we determine if the two circles intersect?

### Tip
Remember, the center of a circle \((x - h)^2 + (y - k)^2 = r^2\) is \((h, k)\), and the radius is \(\sqrt{r^2}\).

Learn how to find the centers of circles given by equations and calculate the distance between these centers. This problem involves using the equation of a circle and the distance formula, suitable for high school students in Grades 9-12.

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