Locus of the mid-points of the chords of the circle `x^(2)+y^(2)=a^(2)` which are always at a constant distance from the centre is

Options

A. parabola
B. ellipse
C. circle
D. none

Answer

To find the locus of the midpoints of the chords of the circle \( x^2 + y^2 = a^2 \) that are always at a constant distance from the center, we can follow these steps:

### Step 1: Understand the Circle
The given equation of the circle is \( x^2 + y^2 = a^2 \). The center of this circle is at the origin (0, 0) and the radius is \( a \).

### Step 2: Define the Midpoint of the Chord
Let the midpoint of the chord be denoted as \( P(h, k) \). We need to find the locus of this point \( P \).

### Step 3: Establish the Constant Distance
According to the problem, the distance from the center of the circle (0, 0) to the midpoint \( P(h, k) \) is a constant \( d \). This distance can be expressed using the distance formula:
\[ \text{Distance} = \sqrt{(h - 0)^2 + (k - 0)^2} = \sqrt{h^2 + k^2} \]
Since this distance is constant, we can set:
\[ \sqrt{h^2 + k^2} = d \]

### Step 4: Square Both Sides
To eliminate the square root, we square both sides of the equation:
\[ h^2 + k^2 = d^2 \]

### Step 5: Substitute Variables
Now, we can replace \( h \) and \( k \) with \( x \) and \( y \) respectively, as they represent the coordinates of the midpoint \( P \):
\[ x^2 + y^2 = d^2 \]

### Step 6: Identify the Locus
The equation \( x^2 + y^2 = d^2 \) represents a circle with a radius of \( d \) centered at the origin (0, 0).

### Conclusion
Thus, the locus of the midpoints of the chords of the circle \( x^2 + y^2 = a^2 \) that are always at a constant distance \( d \) from the center is given by the equation:
\[ x^2 + y^2 = d^2 \]
This means that the locus is also a circle.