If `m(x-2)+sqrt(1-m^2) y= 3` , is tangent to a circle for all `m in [-1, 1]` then the radius of the circle is

Options

A. 1.5
B. 2
C. 4.5
D. 3

Explanation

To solve the problem, we need to determine the radius of the circle for which the given line is tangent for all values of \( m \) in the interval \([-1, 1]\).

### Step-by-Step Solution:

1. **Identify the given equation**:
The equation of the line is given as:
\[ m(x - 2) + \sqrt{1 - m^2} y = 3 \]

2. **Substitute \( m \) with \( \cos \theta \)**:
Let \( m = \cos \theta \). Then the equation becomes:
\[ \cos \theta (x - 2) + \sqrt{1 - \cos^2 \theta} y = 3 \]
Since \( \sqrt{1 - \cos^2 \theta} = \sin \theta \), we can rewrite the equation as:
\[ \cos \theta (x - 2) + \sin \theta y = 3 \]

3. **Rearranging the equation**:
Rearranging gives:
\[ \cos \theta (x - 2) + \sin \theta y - 3 = 0 \]

4. **Form of the tangent line**:
This can be recognized as the equation of a line in the form:
\[ Ax + By + C = 0 \]
where \( A = \cos \theta \), \( B = \sin \theta \), and \( C = -3 + 2\cos \theta \).

5. **Finding the center of the circle**:
The line is tangent to a circle centered at \( (2, 0) \). The general equation of the circle can be expressed as:
\[ (x - 2)^2 + y^2 = r^2 \]
where \( r \) is the radius.

6. **Distance from the center to the line**:
The distance \( d \) from the center \( (2, 0) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|A(2) + B(0) + C|}{\sqrt{A^2 + B^2}} \]
Substituting \( A \) and \( B \):
\[ d = \frac{|\cos \theta (2) + 0 + (-3 + 2\cos \theta)|}{\sqrt{\cos^2 \theta + \sin^2 \theta}} = \frac{|2\cos \theta - 3 + 2\cos \theta|}{1} = |4\cos \theta - 3| \]

7. **Condition for tangency**:
For the line to be tangent to the circle, the distance \( d \) must equal the radius \( r \):
\[ |4\cos \theta - 3| = r \]

8. **Finding the maximum and minimum values**:
We need to find the maximum and minimum values of \( |4\cos \theta - 3| \) as \( \theta \) varies. The expression \( 4\cos \theta - 3 \) varies from:
- When \( \cos \theta = 1 \): \( 4(1) - 3 = 1 \)
- When \( \cos \theta = -1 \): \( 4(-1) - 3 = -7 \)

Therefore, the minimum value is \( |1| = 1 \) and the maximum value is \( |-7| = 7 \).

9. **Conclusion**:
Since the line is tangent for all \( m \) in \([-1, 1]\), the radius must be equal to the maximum distance, which is 3.

Thus, the radius of the circle is:
\[ \boxed{3} \]