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.