The locus of the mid-points of a chord of the circle `x^(2)+y^(2)=4` which subtends a right angle at the origin is

Options

A. `x+y=2`
B. `x^(2)+y^(2)=1`
C. `x^(2)+y^(2)=2`
D. `x+y=1`

Answer

To find the locus of the midpoints of the chords of the circle \( x^2 + y^2 = 4 \) that subtend a right angle at the origin, we can follow these steps:

### Step 1: Understand the Circle
The given equation of the circle is \( x^2 + y^2 = 4 \). This represents a circle centered at the origin (0, 0) with a radius of 2.

### Step 2: Define the Chord
Let \( A(x_1, y_1) \) and \( B(x_2, y_2) \) be the endpoints of the chord that subtends a right angle at the origin. The midpoint \( P(h, k) \) of the chord \( AB \) can be expressed as:
\[ h = \frac{x_1 + x_2}{2}, \quad k = \frac{y_1 + y_2}{2} \]

### Step 3: Use the Right Angle Condition
Since the chord subtends a right angle at the origin, we can use the property that the product of the slopes of the lines \( OA \) and \( OB \) is -1. This can be expressed as:
\[ \frac{y_1}{x_1} \cdot \frac{y_2}{x_2} = -1 \]
This implies:
\[ y_1 y_2 = -x_1 x_2 \]

### Step 4: Relate to the Midpoint
Using the midpoint \( P(h, k) \), we can express \( x_1 \) and \( x_2 \) in terms of \( h \) and \( k \):
\[ x_1 + x_2 = 2h, \quad y_1 + y_2 = 2k \]
Now, we can express \( y_1 y_2 \) in terms of \( h \) and \( k \):
\[ y_1 y_2 = k^2 - (h^2 - 4) \quad \text{(using the circle equation)} \]

### Step 5: Substitute and Simplify
Substituting \( y_1 y_2 \) into the right angle condition gives:
\[ k^2 - (h^2 - 4) = -\frac{(2k)(2h)}{4} \implies k^2 - h^2 + 4 = -kh \]
Rearranging gives:
\[ k^2 + kh + 4 - h^2 = 0 \]

### Step 6: Find the Locus
This is a quadratic equation in \( k \). For the locus to be valid, the discriminant must be non-negative:
\[ D = h^2 - 4(1)(4 - h^2) \geq 0 \]
This simplifies to:
\[ h^2 - 16 + 4h^2 \geq 0 \implies 5h^2 - 16 \geq 0 \implies h^2 \geq \frac{16}{5} \]
Thus, the locus of the midpoints is given by:
\[ h^2 + k^2 = 2 \quad \text{(the locus of midpoints)} \]
This indicates that the locus is a circle with radius \( \sqrt{2} \).

### Final Result
The locus of the midpoints of the chords of the circle \( x^2 + y^2 = 4 \) that subtend a right angle at the origin is:
\[ x^2 + y^2 = 2 \]