PQ is a straight line brawn through O, one of the common points of two circles, and meets them, again in P and Q , find the locus of the point S which bisects the line PQ

Answer

To find the locus of the point S which bisects the line segment PQ, where PQ is a straight line drawn through the common point O of two circles, we can follow these steps:

### Step 1: Understand the Geometry
We have two circles that intersect at point O. A straight line PQ passes through O and intersects the two circles at points P and Q. The point S is the midpoint of the line segment PQ.

### Step 2: Set Up the Coordinate System
Assume the origin O is at (0, 0) for simplicity. Let the centers of the two circles be at points C1 and C2, with radii R1 and R2 respectively.

### Step 3: Define the Points P and Q
Let the points P and Q be represented in polar coordinates relative to the origin O. The angles made by the line with the horizontal axis can be denoted as θ and θ' for points P and Q respectively.

### Step 4: Express Points P and Q
Using polar coordinates:
- The coordinates of point P can be expressed as:
\[ P = (R_1 \cos(\theta + \alpha), R_1 \sin(\theta + \alpha)) \]
- The coordinates of point Q can be expressed as:
\[ Q = (R_2 \cos(\theta + \beta), R_2 \sin(\theta + \beta)) \]
where α and β are the angles corresponding to the points P and Q on their respective circles.

### Step 5: Find the Midpoint S
The coordinates of the midpoint S of line segment PQ can be calculated as:
\[ S = \left( \frac{x_P + x_Q}{2}, \frac{y_P + y_Q}{2} \right) \]
Substituting the coordinates of P and Q:
\[ S = \left( \frac{R_1 \cos(\theta + \alpha) + R_2 \cos(\theta + \beta)}{2}, \frac{R_1 \sin(\theta + \alpha) + R_2 \sin(\theta + \beta)}{2} \right) \]

### Step 6: Determine the Locus of S
To find the locus of point S as the line PQ rotates around point O, we need to express S in a form that eliminates θ. This can be achieved by using trigonometric identities and recognizing that as θ varies, S traces out a circle.

### Step 7: Conclude the Locus
The locus of point S, which is the midpoint of line segment PQ, will be a circle centered at O with a radius that depends on the radii of the two circles and the angles α and β. The exact radius can be derived from the expressions obtained in the previous steps.

### Final Result
The locus of point S is a circle centered at the origin O.

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