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The image contains a series of math problems related to circles, including finding equations of circles, chords, and tangents. I'll go through one of the problems and provide a detailed solution.

Let's start with **Problem 3**:

**Problem 3:**
> Find the equation of the circle touching the line \( y = 2x + 5 \) and touching the line \( y = 2x + 1 \) in the first quadrant.

### Solution:
To find the equation of the circle, we need to determine its center and radius.

1. **Equation of the line:** The line \( y = 2x + 5 \) has a slope of 2, and similarly, the line \( y = 2x + 1 \) has the same slope, indicating that the lines are parallel.

2. **Distance between parallel lines:**
   The distance between two parallel lines \( ax + by + c_1 = 0 \) and \( ax + by + c_2 = 0 \) is given by:
   \[
   \text{Distance} = \frac{|c_2 - c_1|}{\sqrt{a^2 + b^2}}
   \]
   Here, the equations are:
   \[
   y - 2x - 5 = 0 \quad \text{and} \quad y - 2x - 1 = 0
   \]
   So, the distance between these two lines is:
   \[
   \text{Distance} = \frac{|(-1) - (-5)|}{\sqrt{1^2 + (-2)^2}} = \frac{4}{\sqrt{5}}
   \]

3. **Radius of the circle:**
   The circle touches both lines, so the radius \( r \) of the circle is half the distance between these two lines:
   \[
   r = \frac{2}{\sqrt{5}}
   \]

4. **Center of the circle:**
   The center \( (h, k) \) lies on the angle bisector of the lines \( y = 2x + 5 \) and \( y = 2x + 1 \). 
   The equation of the angle bisector can be found using the formula for the angle bisector of two lines \( ax + by + c_1 = 0 \) and \( ax + by + c_2 = 0 \):
   \[
   \frac{|y - 2x - 5|}{\sqrt{1^2 + (-2)^2}} = \frac{|y - 2x - 1|}{\sqrt{1^2 + (-2)^2}}
   \]
   Simplifying this, the equation of the angle bisector is:
   \[
   y = 2x + 3
   \]
   Therefore, the center \( (h, k) \) lies on this line.

5. **Final equation of the circle:**
   The general equation of the circle is:
   \[
   (x - h)^2 + (y - k)^2 = r^2
   \]
   We already know that \( h = x \) and \( k = 2x + 3 \). Substituting \( r = \frac{2}{\sqrt{5}} \), the equation of the circle becomes:
   \[
   (x - h)^2 + (y - (2h + 3))^2 = \left(\frac{2}{\sqrt{5}}\right)^2
   \]

   This gives us the equation of the circle in its standard form.

### Do you want more details on this problem, or would you like me to solve another one from the list? 

Here are five questions related to this problem:
1. What is the significance of the angle bisector in finding the center of the circle?
2. How do you calculate the distance between two parallel lines?
3. Why is the radius of the circle half the distance between the lines?
4. Can the center of the circle be determined by any other method?
5. What would be the equation of the circle if the lines had different slopes?

**Tip:** When dealing with circles and lines, always check if the lines are parallel or intersecting, as this greatly affects the strategy to find the circle's equation.

Discover how to find the equation of a circle touching two lines, y = 2x + 5 and y = 2x + 1, in the first quadrant. This problem involves circle geometry and the distance formula, essential for advanced high school students.

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