9 Circle C₁ has centre (-2, 5) and radius 7. Circle C₂ has centre (12,5).
a Given that the two circles are tangent to each other, find the two possible values for the radius of C2.

To solve the problem, let's consider the information provided:

- Circle \( C_1 \) has a center at \((-2, 5)\) and a radius of 7.
- Circle \( C_2 \) has a center at \((12, 5)\) and an unknown radius \( r_2 \).
- The two circles are tangent to each other.

### Tangency Condition

When two circles are tangent to each other, the distance between their centers is equal to the sum or difference of their radii. There are two types of tangency:

1. **Externally Tangent**: The circles touch each other from the outside. In this case, the distance between the centers of the circles equals the sum of their radii.
   
   \[
   \text{Distance between centers} = r_1 + r_2
   \]

2. **Internally Tangent**: One circle is inside the other, and they touch at a single point. In this case, the distance between the centers equals the difference of their radii.
   
   \[
   \text{Distance between centers} = |r_1 - r_2|
   \]

### Step 1: Calculate the distance between the centers

The distance \( d \) between the centers of \( C_1 \) and \( C_2 \) can be calculated using the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting the given center coordinates:

\[
d = \sqrt{(12 - (-2))^2 + (5 - 5)^2}
\]
\[
d = \sqrt{(12 + 2)^2 + 0^2} = \sqrt{14^2} = 14
\]

### Step 2: Set up the equations for the tangent cases

#### Externally Tangent:

For the circles to be externally tangent:

\[
d = r_1 + r_2
\]

Substituting the values:

\[
14 = 7 + r_2
\]

Solving for \( r_2 \):

\[
r_2 = 14 - 7 = 7
\]

#### Internally Tangent:

For the circles to be internally tangent:

\[
d = |r_1 - r_2|
\]

Substituting the values:

\[
14 = |7 - r_2|
\]

This gives two possible equations:

1. \( 14 = 7 - r_2 \) ⟹ \( r_2 = 7 - 14 = -7 \) (not possible since radius cannot be negative)
2. \( 14 = r_2 - 7 \) ⟹ \( r_2 = 14 + 7 = 21 \)

### Conclusion

The two possible values for the radius \( r_2 \) of circle \( C_2 \) are:

- \( r_2 = 7 \) (for external tangency)
- \( r_2 = 21 \) (for internal tangency)

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the equation of circle \( C_1 \) in standard form?
2. How would the scenario change if the circles were not tangent but intersected at two points?
3. Can two circles have more than one tangent line? If so, how many?
4. What would be the radius of \( C_2 \) if the circles were concentric?
5. How can we determine if two circles are externally or internally tangent given their equations?

**Tip:** When dealing with circles in coordinate geometry, always start by identifying key parameters like the center and radius. This simplifies the application of geometric and algebraic principles.

Circle C₁ has centre (-2, 5) and radius 7. Circle C₂ has centre (12, 5). Given that the two circles are tangent to each other, find the two possible values for the radius of C2.

Discover how to find the possible values for the radius of a circle when two circles with given centers are tangent to each other. This problem involves key concepts in coordinate geometry and tangency conditions, suitable for students in Grades 9-12.

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