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.