Two circles of radius 4 units and 3 units are at some distance such that the length of the transverse common tangent and the length of their direct common tangent are in the ratio 1 : 2. What is the distance between the centres of those circles.

Options

A. `sqrt(50)`
B. `sqrt(65)`
C. 8
D. Cannot be determined

Answer

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

### Step 1: Define the Variables
Let:
- \( r_1 = 4 \) units (radius of the first circle)
- \( r_2 = 3 \) units (radius of the second circle)
- \( d \) = distance between the centers of the circles

### Step 2: Write the Formulas for TCT and DCT
The formulas for the lengths of the tangents are:
- Transverse Common Tangent (TCT):
\[ TCT = \sqrt{d^2 - (r_1 + r_2)^2} = \sqrt{d^2 - (4 + 3)^2} = \sqrt{d^2 - 49} \]

- Direct Common Tangent (DCT):
\[ DCT = \sqrt{d^2 - (r_1 - r_2)^2} = \sqrt{d^2 - (4 - 3)^2} = \sqrt{d^2 - 1} \]

### Step 3: Set Up the Ratio
According to the problem, the ratio of TCT to DCT is given as:
\[ \frac{TCT}{DCT} = \frac{1}{2} \]
Substituting the expressions for TCT and DCT:
\[ \frac{\sqrt{d^2 - 49}}{\sqrt{d^2 - 1}} = \frac{1}{2} \]

### Step 4: Cross Multiply
Cross-multiplying gives:
\[ 2\sqrt{d^2 - 49} = \sqrt{d^2 - 1} \]

### Step 5: Square Both Sides
Squaring both sides to eliminate the square roots:
\[ 4(d^2 - 49) = d^2 - 1 \]

### Step 6: Expand and Rearrange
Expanding the left side:
\[ 4d^2 - 196 = d^2 - 1 \]
Rearranging gives:
\[ 4d^2 - d^2 = 196 - 1 \]
\[ 3d^2 = 195 \]

### Step 7: Solve for \( d^2 \)
Dividing both sides by 3:
\[ d^2 = \frac{195}{3} = 65 \]

### Step 8: Find \( d \)
Taking the square root of both sides:
\[ d = \sqrt{65} \]

### Conclusion
The distance between the centers of the circles is \( \sqrt{65} \) units.

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