If a line through P(-2,3) meets the circle `x^(2)+y^(2)-4x+2y+k=0` at A and B such that PA.PB=31 then the radius of the circles is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem step by step, we will follow the given information and apply the properties of circles.

### Step 1: Write the equation of the circle
The given equation of the circle is:
\[ x^2 + y^2 - 4x + 2y + k = 0 \]

### Step 2: Identify the point P
The point P is given as:
\[ P(-2, 3) \]

### Step 3: Substitute point P into the circle equation
To find the length of the tangent from point P to the circle, we substitute the coordinates of P into the circle equation. The formula for the length of the tangent \( PT \) from point \( P(x_1, y_1) \) to the circle is given by:
\[ PT^2 = x_1^2 + y_1^2 - 4x_1 + 2y_1 + k \]

Substituting \( P(-2, 3) \):
\[ PT^2 = (-2)^2 + (3)^2 - 4(-2) + 2(3) + k \]
Calculating each term:
\[ = 4 + 9 + 8 + 6 + k \]
\[ = 27 + k \]

### Step 4: Use the property of the circle
According to the problem, we have:
\[ PA \cdot PB = 31 \]
By the property of tangents, we know:
\[ PT^2 = PA \cdot PB \]
Thus, we can set up the equation:
\[ 27 + k = 31 \]

### Step 5: Solve for k
Rearranging the equation gives:
\[ k = 31 - 27 = 4 \]

### Step 6: Write the updated equation of the circle
Now that we have found \( k \), we can write the equation of the circle as:
\[ x^2 + y^2 - 4x + 2y + 4 = 0 \]

### Step 7: Identify the center and radius of the circle
The general form of the circle is:
\[ x^2 + y^2 + gx + fy + c = 0 \]
From our equation, we have:
- \( g = -4 \)
- \( f = 2 \)
- \( c = 4 \)

The radius \( r \) of the circle can be calculated using the formula:
\[ r = \sqrt{g^2 + f^2 - c} \]
Substituting the values:
\[ r = \sqrt{(-4)^2 + (2)^2 - 4} \]
Calculating:
\[ = \sqrt{16 + 4 - 4} = \sqrt{16} = 4 \]

### Conclusion
The radius of the circle is:
\[ \boxed{4} \]