Find the locus of the mid point of the chord of a circle `x^(2) + y^(2) = 4` such that the segment intercepted by the chord on the curve `x^(2) – 2x – 2y = 0` subtends a right angle at the origin.

Answer

To find the locus of the midpoint of the chord of the circle \( x^2 + y^2 = 4 \) such that the segment intercepted by the chord on the curve \( x^2 - 2x - 2y = 0 \) subtends a right angle at the origin, we can follow these steps:

### Step 1: Understand the conditions
We have a circle defined by the equation \( x^2 + y^2 = 4 \) and a curve given by \( x^2 - 2x - 2y = 0 \). The chord of the circle must subtend a right angle at the origin.

### Step 2: Identify the midpoint of the chord
Let the midpoint of the chord be \( (h, k) \). The equation of the chord of contact from point \( (h, k) \) to the circle \( x^2 + y^2 = 4 \) is given by:
\[ xh + yk = h^2 + k^2 \]

### Step 3: Set up the equation for the curve
The second curve can be rearranged as:
\[ x^2 - 2x - 2y = 0 \implies y = \frac{x^2 - 2x}{2} \]
This implies that the points where the chord intersects the curve must satisfy this equation.

### Step 4: Apply the right angle condition
For the chord to subtend a right angle at the origin, the coefficients of \( x^2 \) and \( y^2 \) in the equation must satisfy:
\[ \text{Coefficient of } x^2 + \text{Coefficient of } y^2 = 0 \]
From the equation of the chord of contact, we can express it as:
\[ \frac{h^2}{h^2 + k^2} + \frac{k^2}{h^2 + k^2} = 0 \]
This leads us to:
\[ h^2 + k^2 = 0 \]
This condition implies that \( h^2 + k^2 = 0 \) cannot hold true for real numbers unless both \( h \) and \( k \) are zero.

### Step 5: Substitute into the curve equation
Now we need to ensure that the chord intersects the curve. The equation of the curve can be rewritten in terms of \( h \) and \( k \):
\[ x^2 - 2x \cdot \frac{xh + yk}{h^2 + k^2} - 2y \cdot \frac{xh + yk}{h^2 + k^2} = 0 \]
This leads us to:
\[ x^2 - 2x \cdot \frac{h}{h^2 + k^2} - 2y \cdot \frac{k}{h^2 + k^2} = 0 \]

### Step 6: Collect coefficients
From the above equation, we can collect coefficients:
- Coefficient of \( x^2 \) is \( 1 \)
- Coefficient of \( y^2 \) is \( -2 \cdot \frac{k}{h^2 + k^2} \)

Setting the sum of coefficients to zero gives:
\[ 1 - 2 \cdot \frac{k}{h^2 + k^2} = 0 \]
This leads to:
\[ h^2 + k^2 - 2k = 0 \]

### Step 7: Final locus equation
Now, substituting \( h \) and \( k \) back into the equation gives:
\[ h^2 + k^2 - 2h - 2k = 0 \]
This can be rearranged to:
\[ x^2 + y^2 - 2x - 2y = 0 \]

### Conclusion
The locus of the midpoint of the chord is:
\[ x^2 + y^2 - 2x - 2y = 0 \]