The length of the transversal common tangent to the circle `x^(2)+y^(2)=1` and `(x-t)^(2)+y^(2)=1` is `sqrt(21)`, then t=

Options

A. `pm2`
B. `pm5`
C. `pm3`
D. none of these

Explanation

To solve the problem, we need to find the value of \( t \) given the length of the transversal common tangent to the circles defined by the equations \( x^2 + y^2 = 1 \) and \( (x - t)^2 + y^2 = 1 \) is \( \sqrt{21} \).

### Step-by-step Solution:

1. **Identify the centers and radii of the circles**:
- The first circle \( x^2 + y^2 = 1 \) has its center at \( (0, 0) \) and radius \( r_1 = 1 \).
- The second circle \( (x - t)^2 + y^2 = 1 \) has its center at \( (t, 0) \) and radius \( r_2 = 1 \).

2. **Calculate the distance \( d \) between the centers**:
- The distance \( d \) between the centers \( (0, 0) \) and \( (t, 0) \) is given by:
\[ d = |t - 0| = |t| \]

3. **Use the formula for the length of the transversal common tangent**:
- The formula for the length \( L \) of the transversal common tangent is:
\[ L = \sqrt{d^2 - (r_1 + r_2)^2} \]
- Here, \( r_1 + r_2 = 1 + 1 = 2 \).

4. **Substitute the known values into the formula**:
- We know \( L = \sqrt{21} \), so we set up the equation:
\[ \sqrt{21} = \sqrt{d^2 - (2)^2} \]
- Squaring both sides gives:
\[ 21 = d^2 - 4 \]

5. **Solve for \( d^2 \)**:
- Rearranging the equation gives:
\[ d^2 = 21 + 4 = 25 \]

6. **Find \( d \)**:
- Since \( d = |t| \), we have:
\[ |t| = \sqrt{25} = 5 \]

7. **Determine the values of \( t \)**:
- This results in two possible values for \( t \):
\[ t = 5 \quad \text{or} \quad t = -5 \]

### Final Answer:
Thus, the values of \( t \) are \( t = 5 \) or \( t = -5 \).

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