Two circles have the following properties:- the x-axis is a common tangent to the circles- the point (8,2) lies on both circles- the centre of each circle lies on the line x+2y=22.a Find the equation of each circle.b Prove that the line 4x+3y=88 is a common tangent to these circles.
radius of cricle1 is 5 and radius of circle2 is 10 show the solution

Given the problem, we need to find the equations of two circles and prove that a specific line is a common tangent to these circles. Here's a step-by-step solution:

### Part (a): Finding the Equation of Each Circle

1. **Properties of the circles:**
   - The x-axis is a common tangent to both circles.
   - The point (8,2) lies on both circles.
   - The center of each circle lies on the line \( x + 2y = 22 \).

2. **Equation of the line where centers lie:**
   \[
   x + 2y = 22
   \]

3. **Centers and Radii of the Circles:**
   - Let's denote the centers of the circles as \( C_1(a, b) \) and \( C_2(a, b) \).
   - Since the x-axis is a common tangent and given the radii, the distances from the centers to the x-axis must be equal to the radii:
     - For circle 1: \( b = 5 \) (radius is 5)
     - For circle 2: \( b = 10 \) (radius is 10)

4. **Finding the x-coordinates of the centers:**
   - Substitute \( b \) values into the line equation \( x + 2y = 22 \):
     - For circle 1: \( a + 2(5) = 22 \)
       \[
       a + 10 = 22 \implies a = 12
       \]
     - For circle 2: \( a + 2(10) = 22 \)
       \[
       a + 20 = 22 \implies a = 2
       \]

5. **Centers of the circles:**
   - Center of circle 1: \( C_1(12, 5) \)
   - Center of circle 2: \( C_2(2, 10) \)

6. **Equation of the circles:**
   - Circle 1: Center \( (12, 5) \), radius 5
     \[
     (x - 12)^2 + (y - 5)^2 = 25
     \]
   - Circle 2: Center \( (2, 10) \), radius 10
     \[
     (x - 2)^2 + (y - 10)^2 = 100
     \]

### Part (b): Prove that the Line \( 4x + 3y = 88 \) is a Common Tangent

1. **Equation of the tangent line:**
   \[
   4x + 3y = 88
   \]

2. **Distance from the center to the line:**
   - The formula for the distance from a point \((x_1, y_1)\) to a line \(Ax + By + C = 0\) is:
     \[
     \text{Distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}
     \]

3. **For Circle 1:**
   - Center \( (12, 5) \)
   - Distance to the line \( 4x + 3y - 88 = 0 \):
     \[
     \text{Distance} = \frac{|4(12) + 3(5) - 88|}{\sqrt{4^2 + 3^2}} = \frac{|48 + 15 - 88|}{5} = \frac{|63 - 88|}{5} = \frac{25}{5} = 5
     \]
   - Since this distance is equal to the radius of circle 1, the line is tangent to circle 1.

4. **For Circle 2:**
   - Center \( (2, 10) \)
   - Distance to the line \( 4x + 3y - 88 = 0 \):
     \[
     \text{Distance} = \frac{|4(2) + 3(10) - 88|}{\sqrt{4^2 + 3^2}} = \frac{|8 + 30 - 88|}{5} = \frac{|38 - 88|}{5} = \frac{50}{5} = 10
     \]
   - Since this distance is equal to the radius of circle 2, the line is tangent to circle 2.

Therefore, the line \( 4x + 3y = 88 \) is a common tangent to both circles.

### Summary
- The equations of the circles are:
  \[
  (x - 12)^2 + (y - 5)^2 = 25
  \]
  \[
  (x - 2)^2 + (y - 10)^2 = 100
  \]
- The line \( 4x + 3y = 88 \) is a common tangent to both circles.

Would you like more details or have any questions?

### Relative Questions

1. How do you find the center of a circle given its equation?
2. What is the general form of the equation of a circle?
3. How do you determine if a line is tangent to a circle?
4. Can two circles have more than one common tangent? Explain.
5. How do you derive the distance formula between a point and a line?
6. What are the properties of circles with a common tangent?
7. How do you find the points of tangency of a tangent line to a circle?
8. What are some applications of tangent lines in real-world problems?

### Tip

When dealing with geometric problems involving circles and tangents, always visualize the problem with a diagram to better understand the relationships between the different elements.

Explore the solution to a circle geometry problem involving common tangents and circle equations. Learn how to find circle centers, derive equations, and prove a line as a common tangent. Suitable for high school students studying geometry and algebra.

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