Two circles of radius 4 and 3 cm are at some distance s.t. length of the transversal common tangent and length of direct common tangent are in the ratio 1 : 2 find the distance b/w centres

Answer

To find the distance between the centers of two circles with radii 4 cm and 3 cm, given that the lengths of the transversal common tangent (TCT) and the direct common tangent (DCT) are in the ratio 1:2, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Variables:**
- Let \( r_1 = 4 \) cm (radius of the first circle).
- Let \( r_2 = 3 \) cm (radius of the second circle).
- Let \( d \) be the distance between the centers of the two circles.

2. **Formulas for Tangents:**
- The length of the transversal common tangent (TCT) is given by:
\[ TCT = \sqrt{d^2 - (r_1 + r_2)^2} \]
- The length of the direct common tangent (DCT) is given by:
\[ DCT = \sqrt{d^2 - (r_1 - r_2)^2} \]

3. **Set Up the Ratio:**
- According to the problem, the ratio of TCT to DCT is \( 1:2 \):
\[ \frac{TCT}{DCT} = \frac{1}{2} \]
- Substituting the formulas, we have:
\[ \frac{\sqrt{d^2 - (r_1 + r_2)^2}}{\sqrt{d^2 - (r_1 - r_2)^2}} = \frac{1}{2} \]

4. **Cross Multiply and Square Both Sides:**
- Cross multiplying gives:
\[ 2\sqrt{d^2 - (r_1 + r_2)^2} = \sqrt{d^2 - (r_1 - r_2)^2} \]
- Squaring both sides results in:
\[ 4(d^2 - (r_1 + r_2)^2) = d^2 - (r_1 - r_2)^2 \]

5. **Substituting the Values of \( r_1 \) and \( r_2 \):**
- Calculate \( r_1 + r_2 = 4 + 3 = 7 \) and \( r_1 - r_2 = 4 - 3 = 1 \).
- Substitute these values into the equation:
\[ 4(d^2 - 7^2) = d^2 - 1^2 \]
- This simplifies to:
\[ 4(d^2 - 49) = d^2 - 1 \]

6. **Expanding and Rearranging:**
- Expanding gives:
\[ 4d^2 - 196 = d^2 - 1 \]
- Rearranging terms results in:
\[ 4d^2 - d^2 = 196 - 1 \]
\[ 3d^2 = 195 \]

7. **Solving for \( d^2 \):**
- Dividing both sides by 3:
\[ d^2 = \frac{195}{3} = 65 \]

8. **Finding \( d \):**
- Taking the square root gives:
\[ d = \sqrt{65} \text{ cm} \]

### Final Answer:
The distance between the centers of the two circles is \( \sqrt{65} \) cm.