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I see that you've uploaded two images with several geometry and circle-related questions. I'll start by solving the first question and work through others. Here’s the solution to the first problem:

### Question 1:
The circle \( C \) has the equation:
\[
x^2 + y^2 - 2x + 6y = 26
\]
#### (i) Coordinates of the center of \( C \)
We start by converting the given equation into standard form \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center and \( r \) is the radius.

1. Group the \( x \)-terms and \( y \)-terms:
   \[
   (x^2 - 2x) + (y^2 + 6y) = 26
   \]

2. Complete the square:
   - For \( x^2 - 2x \): Take half of \( -2 \) (which is \( -1 \)), and square it to get \( 1 \). So, add and subtract 1 inside the equation:
     \[
     (x^2 - 2x + 1 - 1)
     \]
   - For \( y^2 + 6y \): Take half of \( 6 \) (which is \( 3 \)), and square it to get \( 9 \). So, add and subtract 9 inside the equation:
     \[
     (y^2 + 6y + 9 - 9)
     \]

3. Rewriting the equation:
   \[
   (x - 1)^2 - 1 + (y + 3)^2 - 9 = 26
   \]
   Simplify it:
   \[
   (x - 1)^2 + (y + 3)^2 = 36
   \]

4. Now we have the equation of the circle in standard form:
   \[
   (x - 1)^2 + (y + 3)^2 = 36
   \]
   From this, the **center** \( C \) is \( (1, -3) \).

#### (ii) Radius of \( C \)
The radius \( r \) is the square root of 36, so:
\[
r = \sqrt{36} = 6
\]

### Final Answer:
- Center: \( (1, -3) \)
- Radius: \( 6 \)

Let me know if you'd like details for the next question or any clarification. Here are 5 related questions to deepen understanding:

1. What is the general form of a circle’s equation?
2. How do you complete the square for a quadratic expression?
3. How can you find the distance between two points using their coordinates?
4. What changes in the equation of a circle when it is translated on the coordinate plane?
5. How can you determine if a point lies on a given circle?

### Tip:
Always remember to complete the square carefully when converting the general form of a circle’s equation to standard form.

Find the center and radius of the circle based on the equation x^2 + y^2 - 2x + 6y = 26, as well as additional related questions on circle geometry.

Step-by-step solutions for finding the center and radius of a circle from its general equation, using methods such as completing the square. Learn to find tangents and other geometric properties related to circles, ideal for students in Grades 9-12.

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