The radius of two circles is 3 cm and 4 cm. The distance between the centres of the circles is 10 cm. What is the ratio of the length of direct common tangent to the length of the transverse common tangent?

Options

A. `sqrt(51) : sqrt(68)`
B. `sqrt(33) : sqrt(17)`
C. `sqrt(66) : sqrt(51)`
D. `sqrt(28) : sqrt(17)`

Answer

To find the ratio of the length of the direct common tangent (DCT) to the length of the transverse common tangent (TCT) between two circles, we can use the following formulas:

1. **Direct Common Tangent (DCT)**:
\[ DCT = \sqrt{d^2 - (r_1 - r_2)^2} \]

2. **Transverse Common Tangent (TCT)**:
\[ TCT = \sqrt{d^2 - (r_1 + r_2)^2} \]

Where:
- \(d\) is the distance between the centers of the circles,
- \(r_1\) is the radius of the first circle,
- \(r_2\) is the radius of the second circle.

### Step-by-Step Solution:

**Step 1: Identify the values.**
- Given \(r_1 = 3 \, \text{cm}\), \(r_2 = 4 \, \text{cm}\), and \(d = 10 \, \text{cm}\).

**Step 2: Calculate the Direct Common Tangent (DCT).**
- Substitute the values into the DCT formula:
\[ DCT = \sqrt{d^2 - (r_1 - r_2)^2} \]
\[ DCT = \sqrt{10^2 - (3 - 4)^2} \]
\[ DCT = \sqrt{100 - (-1)^2} \]
\[ DCT = \sqrt{100 - 1} = \sqrt{99} \]

**Step 3: Calculate the Transverse Common Tangent (TCT).**
- Substitute the values into the TCT formula:
\[ TCT = \sqrt{d^2 - (r_1 + r_2)^2} \]
\[ TCT = \sqrt{10^2 - (3 + 4)^2} \]
\[ TCT = \sqrt{100 - 7^2} \]
\[ TCT = \sqrt{100 - 49} = \sqrt{51} \]

**Step 4: Calculate the ratio of DCT to TCT.**
- Now, we find the ratio:
\[ \text{Ratio} = \frac{DCT}{TCT} = \frac{\sqrt{99}}{\sqrt{51}} = \sqrt{\frac{99}{51}} = \sqrt{\frac{33}{17}} \]

### Final Answer:
The ratio of the length of the direct common tangent to the length of the transverse common tangent is:
\[ \sqrt{\frac{33}{17}} \]