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### **Question 1**
The circle \( C \) has the equation:
\[
x^2 + y^2 - 2x + 6y = 26
\]
We need to:
1. **Find the coordinates of the center of C.**
2. **Find the radius of C.**

#### Step 1: Rearranging the equation into standard form
We will complete the square for both \( x \) and \( y \).

- For \( x \), we have \( x^2 - 2x \). Completing the square:
  \[
  x^2 - 2x = (x - 1)^2 - 1
  \]
  
- For \( y \), we have \( y^2 + 6y \). Completing the square:
  \[
  y^2 + 6y = (y + 3)^2 - 9
  \]

Now, rewrite the equation with these squared terms:
\[
(x - 1)^2 - 1 + (y + 3)^2 - 9 = 26
\]
Simplify:
\[
(x - 1)^2 + (y + 3)^2 = 26 + 1 + 9
\]
\[
(x - 1)^2 + (y + 3)^2 = 36
\]

#### Step 2: Finding the center and radius
- The standard form of a circle is \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center and \( r \) is the radius.
- Comparing this with \( (x - 1)^2 + (y + 3)^2 = 36 \), we can see that:
  - The center of the circle is \( (1, -3) \).
  - The radius \( r \) is \( \sqrt{36} = 6 \).

**Answer:**
- The coordinates of the center are \( (1, -3) \).
- The radius is \( 6 \).

---

### **Question 2**
The circle \( C \) has center \( (2, 5) \) and passes through the point \( (4, 9) \). Find the equation for \( C \).

#### Step 1: Use the formula of a circle
The standard equation of a circle is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \( (h, k) \) is the center and \( r \) is the radius.

#### Step 2: Finding the radius
We are given the center \( (2, 5) \) and a point on the circle \( (4, 9) \). The radius is the distance between these two points:
\[
r = \sqrt{(4 - 2)^2 + (9 - 5)^2} = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]

#### Step 3: Writing the equation of the circle
The equation of the circle is:
\[
(x - 2)^2 + (y - 5)^2 = (2\sqrt{5})^2
\]
\[
(x - 2)^2 + (y - 5)^2 = 20
\]

**Answer:** The equation of the circle is \( (x - 2)^2 + (y - 5)^2 = 20 \).

---

### **Question 3**
The circle \( C \) has center \( (-2, 3) \) and passes through point \( (1, 8) \).

1. **(a) Find the equation for C.**

#### Step 1: Use the formula of a circle
The center is \( (-2, 3) \) and the radius can be found by calculating the distance from \( (-2, 3) \) to \( (1, 8) \):
\[
r = \sqrt{(1 - (-2))^2 + (8 - 3)^2} = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34}
\]

#### Step 2: Writing the equation of the circle
The equation of the circle is:
\[
(x + 2)^2 + (y - 3)^2 = (\sqrt{34})^2
\]
\[
(x + 2)^2 + (y - 3)^2 = 34
\]

**Answer:** The equation of the circle is \( (x + 2)^2 + (y - 3)^2 = 34 \).

2. **(b) Show that the point \( (3, 6) \) lies on C.**

Substitute \( x = 3 \) and \( y = 6 \) into the equation:
\[
(3 + 2)^2 + (6 - 3)^2 = 34
\]
\[
5^2 + 3^2 = 25 + 9 = 34
\]
Since both sides are equal, the point \( (3, 6) \) lies on the circle.

---

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The image contains five different circle-related math questions involving equations of circles, finding centers, radii, and tangents.

Learn to solve circle geometry problems involving the equation of a circle, center, radius, tangents, and points on the circle. Detailed step-by-step solutions with key formulas such as the equation of a circle and distance formula. Suitable for high school students.

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