Find the locus of the mid point of the circle `x^2+y^2=a^2` which subtend a right angle at the point (p,q)

Answer

To find the locus of the midpoint of a circle \(x^2 + y^2 = a^2\) that subtends a right angle at the point \((p, q)\), we can follow these steps:

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

**Hint:** Recall that the center of the circle is at the origin and the radius is given as \(a\).

### Step 2: Define the Midpoint
Let the midpoint of the chord of the circle that subtends a right angle at the point \((p, q)\) be denoted as \((h, k)\).

**Hint:** The midpoint is the average of the endpoints of the chord.

### Step 3: Use the Right Angle Condition
For a circle, if a chord subtends a right angle at a point outside the circle, the distance from the center of the circle to the point is equal to the radius of the circle. Thus, we can use the right triangle property.

### Step 4: Apply the Pythagorean Theorem
Using the Pythagorean theorem in triangle \(OAC\) (where \(O\) is the origin, \(A\) is the midpoint, and \(C\) is a point on the circle), we have:
\[ OC^2 = OA^2 + AC^2 \]
Where:
- \(OC = a\) (radius)
- \(OA = \sqrt{h^2 + k^2}\) (distance from origin to midpoint)
- \(AC\) is the distance from the midpoint to the point on the circle.

### Step 5: Calculate Distances
From the midpoint \((h, k)\) to the point \((p, q)\), we have:
\[ AP = \sqrt{(h - p)^2 + (k - q)^2} \]
And since \(C\) lies on the circle, we can express \(AC\) as:
\[ AC = \sqrt{a^2 - (h^2 + k^2)} \]

### Step 6: Set Up the Equation
Using the relationship from the right triangle:
\[ AP^2 + AC^2 = OC^2 \]
Substituting the distances:
\[ (h - p)^2 + (k - q)^2 + (a^2 - (h^2 + k^2)) = a^2 \]

### Step 7: Simplify the Equation
Expanding and simplifying the equation:
\[ (h - p)^2 + (k - q)^2 = h^2 + k^2 \]
This leads to:
\[ h^2 - 2ph + p^2 + k^2 - 2qk + q^2 = h^2 + k^2 \]
Cancelling \(h^2 + k^2\) from both sides gives:
\[ -2ph - 2qk + p^2 + q^2 = 0 \]

### Step 8: Rearrange to Find the Locus
Rearranging the equation results in:
\[ 2h + 2k = p^2 + q^2 \]
Thus, the locus of the midpoint \((h, k)\) can be expressed as:
\[ h + k = \frac{p^2 + q^2}{2} \]

### Final Equation
The locus of the midpoint of the circle that subtends a right angle at the point \((p, q)\) is given by:
\[ h + k = \frac{p^2 + q^2}{2} \]