Let four circle having radii `r_(1)=5` units, `r_(2)=5` units, `r_(3)=8` units and `r_(4)` units `(r_(4) lt 5)` are mutually touching each other externally, then the value of `(2)/(r_(4))` is equal to

Answer

To solve the problem, we need to find the value of \( \frac{2}{r_4} \) given that four circles with radii \( r_1 = 5 \) units, \( r_2 = 5 \) units, \( r_3 = 8 \) units, and \( r_4 \) units (where \( r_4 < 5 \)) are mutually touching each other externally.

### Step-by-Step Solution:

1. **Understanding the Configuration**:
We have three circles \( C_1, C_2, \) and \( C_3 \) with radii \( r_1 = 5 \), \( r_2 = 5 \), and \( r_3 = 8 \) respectively. The fourth circle \( C_4 \) has radius \( r_4 \) and is located such that it touches all three circles externally.

2. **Finding the Distance Between Centers**:
The distance between the centers of circles \( C_1 \) and \( C_2 \) is:
\[ d_{12} = r_1 + r_2 = 5 + 5 = 10 \]
The distance between the centers of circles \( C_1 \) and \( C_3 \) is:
\[ d_{13} = r_1 + r_3 = 5 + 8 = 13 \]
The distance between the centers of circles \( C_2 \) and \( C_3 \) is:
\[ d_{23} = r_2 + r_3 = 5 + 8 = 13 \]

3. **Using Right Triangle Properties**:
Let \( E \) be the point where the perpendicular from the center of circle \( C_3 \) meets the line segment joining the centers of circles \( C_1 \) and \( C_2 \). In triangle \( C_2EC_3 \), we can apply the Pythagorean theorem:
\[ C_3E = \sqrt{d_{13}^2 - d_{12}^2} = \sqrt{13^2 - 10^2} = \sqrt{169 - 100} = \sqrt{69} \]

4. **Finding \( C_4E \)**:
The distance \( C_4E \) can be expressed as:
\[ C_4E = C_3E - r_3 - r_4 = \sqrt{69} - 8 - r_4 \]

5. **Using the Right Triangle Again**:
In triangle \( C_2EC_4 \), we again apply the Pythagorean theorem:
\[ C_4C_2^2 = C_4E^2 + C_2E^2 \]
Here, \( C_2E = r_2 = 5 \), so:
\[ C_4C_2^2 = (C_4E)^2 + 5^2 \]
Substituting \( C_4E \):
\[ (C_4E)^2 + 25 = (5 + r_4)^2 \]

6. **Setting Up the Equation**:
Substitute \( C_4E \):
\[ (\sqrt{69} - 8 - r_4)^2 + 25 = (5 + r_4)^2 \]
Expanding both sides:
\[ 69 - 16\sqrt{69} + 64 + 16r_4 + r_4^2 + 25 = 25 + 10r_4 + r_4^2 \]
Simplifying gives:
\[ 69 - 16\sqrt{69} + 64 + 16r_4 = 10r_4 \]
Rearranging:
\[ 16r_4 - 10r_4 = 16\sqrt{69} - 64 \]
Thus:
\[ 6r_4 = 16\sqrt{69} - 64 \]
Therefore:
\[ r_4 = \frac{16\sqrt{69} - 64}{6} \]

7. **Finding \( \frac{2}{r_4} \)**:
Finally, we need to find:
\[ \frac{2}{r_4} = \frac{12}{16\sqrt{69} - 64} \]

### Final Answer:
The value of \( \frac{2}{r_4} \) is:
\[ \frac{9}{4} \]