If the length of direct common tangent and transverse common tangent of two circles with integral radii are 3 units and 1 unit respectively, then the reciprocal of the square of the distance between the centres of the circles is equal to

Answer

To solve the problem, we need to find the reciprocal of the square of the distance between the centers of two circles given the lengths of their direct and transverse common tangents.

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- Let the radii of the two circles be \( R_1 \) and \( R_2 \).
- The length of the direct common tangent is given as \( 3 \) units.
- The length of the transverse common tangent is given as \( 1 \) unit.

2. **Using the Formulas for Tangents:**
- The formula for the length of the direct common tangent \( L_d \) is:
\[ L_d = \sqrt{d^2 - (R_1 - R_2)^2} \]
- The formula for the length of the transverse common tangent \( L_t \) is:
\[ L_t = \sqrt{d^2 - (R_1 + R_2)^2} \]
- Here, \( d \) is the distance between the centers of the circles.

3. **Setting Up the Equations:**
- From the direct common tangent:
\[ \sqrt{d^2 - (R_1 - R_2)^2} = 3 \]
Squaring both sides:
\[ d^2 - (R_1 - R_2)^2 = 9 \quad \text{(Equation 1)} \]

- From the transverse common tangent:
\[ \sqrt{d^2 - (R_1 + R_2)^2} = 1 \]
Squaring both sides:
\[ d^2 - (R_1 + R_2)^2 = 1 \quad \text{(Equation 2)} \]

4. **Subtracting the Two Equations:**
- Subtract Equation 2 from Equation 1:
\[ (d^2 - (R_1 - R_2)^2) - (d^2 - (R_1 + R_2)^2) = 9 - 1 \]
- This simplifies to:
\[ (R_1 + R_2)^2 - (R_1 - R_2)^2 = 8 \]

5. **Expanding the Squares:**
- Expanding both squares:
\[ (R_1^2 + 2R_1R_2 + R_2^2) - (R_1^2 - 2R_1R_2 + R_2^2) = 8 \]
- Simplifying gives:
\[ 4R_1R_2 = 8 \]
- Thus:
\[ R_1R_2 = 2 \]

6. **Finding Possible Integral Values for Radii:**
- The integral pairs \((R_1, R_2)\) that satisfy \( R_1R_2 = 2 \) are:
- \( (1, 2) \)
- \( (2, 1) \)

7. **Substituting Back to Find \( d^2 \):**
- Let's use \( R_1 = 1 \) and \( R_2 = 2 \):
- Substitute into Equation 1:
\[ d^2 - (1 - 2)^2 = 9 \]
\[ d^2 - 1 = 9 \]
\[ d^2 = 10 \]

8. **Finding the Reciprocal of the Square of the Distance:**
- We need to find \( \frac{1}{d^2} \):
\[ \frac{1}{d^2} = \frac{1}{10} \]

### Final Answer:
The reciprocal of the square of the distance between the centers of the circles is \( \frac{1}{10} \).