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Let's begin by solving **Question 6 part (b)** from the image:

### Problem (6b):
We are given two circles:

1. **Circle \( C_1 \)** has its diameter endpoints at \((-3, -5)\) and \((7, 3)\).
2. **Circle \( C_2 \)** has its center at \((10, 5)\) and the same radius as circle \( C_1 \).

We are tasked to find the equation of circle \( C_2 \).

### Solution for 6(b):

#### Step 1: Find the radius of Circle \( C_1 \)

The radius of \( C_1 \) is half the length of its diameter. The diameter's length is the distance between the points \((-3, -5)\) and \((7, 3)\).

Using the distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\):

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

Substituting the coordinates of the endpoints:

\[
d = \sqrt{(7 - (-3))^2 + (3 - (-5))^2} = \sqrt{(7 + 3)^2 + (3 + 5)^2} = \sqrt{10^2 + 8^2} = \sqrt{100 + 64} = \sqrt{164}
\]

The radius \( r_1 \) of circle \( C_1 \) is half the diameter:

\[
r_1 = \frac{\sqrt{164}}{2} = \frac{2\sqrt{41}}{2} = \sqrt{41}
\]

So, the radius of \( C_1 \) is \( \sqrt{41} \).

#### Step 2: Equation of Circle \( C_2 \)

Circle \( C_2 \) has its center at \((10, 5)\) and the same radius \( r = \sqrt{41} \). The general equation of a circle is given by:

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

where \((h, k)\) is the center and \(r\) is the radius.

For circle \( C_2 \), the center is \((10, 5)\) and the radius is \( \sqrt{41} \), so we substitute into the equation:

\[
(x - 10)^2 + (y - 5)^2 = (\sqrt{41})^2
\]

Simplifying:

\[
(x - 10)^2 + (y - 5)^2 = 41
\]

Thus, the equation of circle \( C_2 \) is:

\[
(x - 10)^2 + (y - 5)^2 = 41
\]

This completes part (b) of the question.

---

Would you like to proceed with part (c), or do you have any questions?  
Here are some related questions:

1. How do you find the equation of a circle given its center and radius?
2. How do you calculate the distance between two points?
3. How do you find the midpoint of a line segment, which is useful in finding the center of a circle from its diameter?
4. What is the relationship between the diameter and radius of a circle?
5. How do you solve problems involving intersections of circles?

**Tip:** Always start by identifying the center and radius of a circle when forming its equation.

Learn how to solve a geometry problem involving the equation of a circle, given the center and radius. This detailed solution involves calculating the distance between two points, using the circle equation formula, and understanding key geometric concepts related to circles. Suitable for students in Grades 10-12.

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