The locus of midpoints of the chord of the circle `x^(2)+y^(2)=25` which pass through a fixed point (4,6) is a circle. The radius of that circle is

Options

A. `sqrt(52)`
B. `sqrt(2)`
C. `sqrt(13)`
D. `sqrt(10)`

Answer

To solve the problem step by step, we will find the locus of the midpoints of the chords of the circle defined by the equation \(x^2 + y^2 = 25\) that pass through the fixed point (4, 6).

### Step 1: Understand the Circle Equation
The given circle has the equation:
\[ x^2 + y^2 = 25 \]
This represents a circle centered at the origin (0,0) with a radius of 5.

**Hint:** Identify the center and radius of the circle from its equation.

### Step 2: Define the Midpoint of the Chord
Let the midpoint of the chord be denoted as \( (H, K) \).

**Hint:** Recognize that the midpoint of a chord can be represented by coordinates \( (H, K) \).

### Step 3: Use the Chord Midpoint Theorem
According to the chord midpoint theorem, the equation for the midpoint of the chord is given by:
\[ T = S_1 \]
where \( T \) is the equation derived from the midpoint and \( S_1 \) is the equation of the circle evaluated at the midpoint.

**Hint:** Recall that \( T \) is derived from the coordinates of the midpoint.

### Step 4: Write the Equation for \( T \)
The equation for \( T \) becomes:
\[ Hx + Ky - 25 = 0 \]

**Hint:** Substitute the midpoint coordinates into the equation.

### Step 5: Write the Equation for \( S_1 \)
The equation for \( S_1 \) is obtained by substituting \( H \) and \( K \) into the circle's equation:
\[ H^2 + K^2 - 25 = 0 \]

**Hint:** This equation represents the condition that the point lies on the circle.

### Step 6: Set \( T = S_1 \)
Now, we equate \( T \) and \( S_1 \):
\[ Hx + Ky - 25 = H^2 + K^2 - 25 \]
This simplifies to:
\[ Hx + Ky = H^2 + K^2 \]

**Hint:** This step involves equating the two expressions derived from the chord midpoint theorem.

### Step 7: Substitute the Fixed Point
Since the chord passes through the fixed point (4, 6), we substitute \( x = 4 \) and \( y = 6 \):
\[ 4H + 6K = H^2 + K^2 \]

**Hint:** Use the coordinates of the fixed point to find the relationship between \( H \) and \( K \).

### Step 8: Rearrange the Equation
Rearranging gives:
\[ H^2 - 4H + K^2 - 6K = 0 \]

**Hint:** This is a standard form of a circle equation.

### Step 9: Complete the Square
To convert this into the standard circle form, we complete the square for both \( H \) and \( K \):
\[ (H^2 - 4H + 4) + (K^2 - 6K + 9) = 13 \]
This simplifies to:
\[ (H - 2)^2 + (K - 3)^2 = 13 \]

**Hint:** Completing the square helps in identifying the center and radius of the circle.

### Step 10: Identify the Radius
From the equation \( (H - 2)^2 + (K - 3)^2 = 13 \), we see that the center of the circle is at \( (2, 3) \) and the radius \( r \) is:
\[ r = \sqrt{13} \]

**Hint:** The radius is the square root of the constant term on the right side of the equation.

### Final Answer
The radius of the locus of midpoints of the chords of the circle that pass through the fixed point (4, 6) is:
\[ \sqrt{13} \]