The equation to the locus of the midpoints of chords of the circle `x^(2)+y^(2)=r^(2)` having a constant length 2l is

Options

A. `x^(2)+y^(2)=l^(2)-r^(2)`
B. `x^(2)+y^(2)=r^(2)-l^(2)`
C. `x^(2)+y^(2)=4l^(2)`
D. `x^(2)+y^(2)=l^(2)+r^(2)`

Answer

To find the equation of the locus of the midpoints of chords of the circle \( x^2 + y^2 = r^2 \) that have a constant length \( 2l \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Circle and Chord**:
- The given circle has the equation \( x^2 + y^2 = r^2 \), with center at the origin \( O(0, 0) \) and radius \( r \).
- Let \( A \) and \( B \) be the endpoints of a chord of the circle, and let \( M(h, k) \) be the midpoint of this chord.

2. **Length of the Chord**:
- The length of the chord \( AB \) is given as \( 2l \). Therefore, the distance from the midpoint \( M \) to either endpoint \( A \) or \( B \) is \( MB = l \).

3. **Using Pythagorean Theorem**:
- In triangle \( OMB \), we can apply the Pythagorean theorem:
\[ OM^2 + MB^2 = OB^2 \]
- Here, \( OM \) is the distance from the center \( O \) to the midpoint \( M \), \( MB \) is the distance from \( M \) to \( B \), and \( OB \) is the radius \( r \).

4. **Expressing Distances**:
- The distance \( OM \) can be expressed as:
\[ OM = \sqrt{h^2 + k^2} \]
- Since \( MB = l \), we have:
\[ MB^2 = l^2 \]
- The distance \( OB \) is simply the radius \( r \), so:
\[ OB^2 = r^2 \]

5. **Setting Up the Equation**:
- Substituting these into the Pythagorean theorem gives:
\[ OM^2 + MB^2 = OB^2 \]
\[ h^2 + k^2 + l^2 = r^2 \]

6. **Rearranging the Equation**:
- Rearranging the equation, we get:
\[ h^2 + k^2 = r^2 - l^2 \]

7. **Substituting Variables**:
- Replace \( h \) with \( x \) and \( k \) with \( y \):
\[ x^2 + y^2 = r^2 - l^2 \]

8. **Final Equation**:
- Thus, the equation of the locus of the midpoints of the chords is:
\[ x^2 + y^2 = r^2 - l^2 \]

### Conclusion:
The equation to the locus of the midpoints of chords of the circle \( x^2 + y^2 = r^2 \) having a constant length \( 2l \) is:
\[ x^2 + y^2 = r^2 - l^2 \]