If the radii of two circles be 6 cm and 3 cm and the length of the transverse common tangent be 8 cm, then the distance be tween the two centres is

Options

A. `sqrt(145) cm `
B. `sqrt(140) cm `
C. `sqrt(150) cm `
D. `sqrt(135) cm `

Answer

To find the distance between the centers of the two circles given their radii and the length of the transverse common tangent, we can use the formula for the length of the transverse common tangent:

\[ L = 2 \sqrt{d^2 - (r_1 + r_2)^2} \]

where:
- \(L\) is the length of the transverse common tangent,
- \(d\) is the distance between the centers of the two circles,
- \(r_1\) and \(r_2\) are the radii of the circles.

Given:
- \(r_1 = 6 \, \text{cm}\)
- \(r_2 = 3 \, \text{cm}\)
- \(L = 8 \, \text{cm}\)

### Step 1: Substitute the values into the formula
We start by substituting the known values into the formula:

\[ 8 = 2 \sqrt{d^2 - (6 + 3)^2} \]

### Step 2: Simplify the equation
Calculate \(r_1 + r_2\):

\[ r_1 + r_2 = 6 + 3 = 9 \]

Now, substitute this back into the equation:

\[ 8 = 2 \sqrt{d^2 - 9^2} \]

### Step 3: Square both sides to eliminate the square root
Square both sides to remove the square root:

\[ 8^2 = (2 \sqrt{d^2 - 81})^2 \]

This simplifies to:

\[ 64 = 4(d^2 - 81) \]

### Step 4: Expand and rearrange the equation
Expand the right side:

\[ 64 = 4d^2 - 324 \]

Now, add 324 to both sides:

\[ 64 + 324 = 4d^2 \]

\[ 388 = 4d^2 \]

### Step 5: Solve for \(d^2\)
Divide both sides by 4:

\[ d^2 = \frac{388}{4} = 97 \]

### Step 6: Find \(d\) by taking the square root
Now take the square root of both sides to find \(d\):

\[ d = \sqrt{97} \]

### Final Answer
The distance between the two centers is:

\[ d \approx 9.85 \, \text{cm} \]