Find the locus of the midpoint of the chords of the circle `x^2+y^2=a^2` which subtend a right angle at the point `(0,0)dot`

Options

A. `x^(2)+y^(2)=(a^(2))/(2)`
B. `x^(2)+y^(2)=2a^(2)`
C. `x^(2)+y^(2)=(a^(2))/(4)`
D. none of these

Explanation

To find the locus of the midpoint of the chords of the circle \(x^2 + y^2 = a^2\) that subtend a right angle at the origin \((0,0)\), we can follow these steps:

### Step 1: Understand the Circle and Chords
The given circle is centered at the origin with radius \(a\). The equation of the circle is:
\[ x^2 + y^2 = a^2 \]
We need to find the locus of the midpoints of chords that subtend a right angle at the origin.

### Step 2: Midpoint of the Chord
Let the midpoint of the chord be denoted as \(D(h, k)\). Since \(D\) is the midpoint of the chord \(AB\), we can denote the endpoints of the chord as \(A(x_1, y_1)\) and \(B(x_2, y_2)\).

### Step 3: Relationship Between Points
Since \(D\) is the midpoint, we have:
\[ h = \frac{x_1 + x_2}{2}, \quad k = \frac{y_1 + y_2}{2} \]
The chord \(AB\) subtends a right angle at the origin, which means that the vectors \(OA\) and \(OB\) are perpendicular. Therefore, we have:
\[ x_1 x_2 + y_1 y_2 = 0 \]

### Step 4: Express \(x_1\) and \(y_1\) in Terms of \(h\) and \(k\)
From the midpoint formula, we can express \(x_1\) and \(x_2\) as:
\[ x_1 = h + p, \quad x_2 = h - p \]
\[ y_1 = k + q, \quad y_2 = k - q \]
where \(p\) and \(q\) are the distances from the midpoint to the endpoints of the chord.

### Step 5: Substitute into the Perpendicular Condition
Substituting these into the perpendicular condition:
\[ (h + p)(h - p) + (k + q)(k - q) = 0 \]
This simplifies to:
\[ h^2 - p^2 + k^2 - q^2 = 0 \]
or
\[ h^2 + k^2 = p^2 + q^2 \]

### Step 6: Use the Circle Equation
Since points \(A\) and \(B\) lie on the circle, we can use the circle's equation:
\[ (h + p)^2 + (k + q)^2 = a^2 \]
and
\[ (h - p)^2 + (k - q)^2 = a^2 \]

### Step 7: Simplify and Find the Locus
By simplifying these equations, we can find that:
\[ h^2 + k^2 + 2hp + 2kq = a^2 \]
and
\[ h^2 + k^2 - 2hp - 2kq = a^2 \]
Subtracting these two equations gives:
\[ 4hp + 4kq = 0 \]
Thus, \(hp + kq = 0\).

### Step 8: Final Locus Equation
From the relationship \(h^2 + k^2 = p^2 + q^2\) and substituting back, we find that the locus of the midpoint \(D(h, k)\) is:
\[ h^2 + k^2 = \frac{a^2}{2} \]
Replacing \(h\) and \(k\) with \(x\) and \(y\), we have:
\[ x^2 + y^2 = \frac{a^2}{2} \]

### Conclusion
The locus of the midpoint of the chords of the circle \(x^2 + y^2 = a^2\) that subtend a right angle at the origin is:
\[ \boxed{x^2 + y^2 = \frac{a^2}{2}} \]