Find the polar equation to the circle described on the straight line joining the points ` (a , alpha ) and (b , beta )` as diameter

Answer

To find the polar equation of the circle described on the straight line joining the points \( (a, \alpha) \) and \( (b, \beta) \) as the diameter, we can follow these steps:

### Step 1: Convert Polar Coordinates to Cartesian Coordinates
The points given in polar coordinates are:
- Point A: \( (a, \alpha) \)
- Point B: \( (b, \beta) \)

We convert these points to Cartesian coordinates using the formulas:
- \( x = r \cos(\theta) \)
- \( y = r \sin(\theta) \)

For Point A:
\[ x_1 = a \cos(\alpha), \quad y_1 = a \sin(\alpha) \]

For Point B:
\[ x_2 = b \cos(\beta), \quad y_2 = b \sin(\beta) \]

### Step 2: Write the Equation of the Circle
The general equation of a circle with diameter endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ (x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0 \]

Substituting the Cartesian coordinates of points A and B:
\[ (x - a \cos(\alpha))(x - b \cos(\beta)) + (y - a \sin(\alpha))(y - b \sin(\beta)) = 0 \]

### Step 3: Substitute Polar Coordinates
In polar coordinates, we have:
- \( x = r \cos(\theta) \)
- \( y = r \sin(\theta) \)

Substituting these into the circle equation:
\[ (r \cos(\theta) - a \cos(\alpha))(r \cos(\theta) - b \cos(\beta)) + (r \sin(\theta) - a \sin(\alpha))(r \sin(\theta) - b \sin(\beta)) = 0 \]

### Step 4: Expand the Equation
Now we expand the equation:
1. For the \( x \) terms:
\[ (r \cos(\theta) - a \cos(\alpha))(r \cos(\theta) - b \cos(\beta)) = r^2 \cos^2(\theta) - (a \cos(\alpha) + b \cos(\beta)) r \cos(\theta) + a b \cos(\alpha) \cos(\beta) \]

2. For the \( y \) terms:
\[ (r \sin(\theta) - a \sin(\alpha))(r \sin(\theta) - b \sin(\beta)) = r^2 \sin^2(\theta) - (a \sin(\alpha) + b \sin(\beta)) r \sin(\theta) + a b \sin(\alpha) \sin(\beta) \]

Combining these, we get:
\[ r^2 (\cos^2(\theta) + \sin^2(\theta)) - r \left[(a \cos(\alpha) + b \cos(\beta)) \cos(\theta) + (a \sin(\alpha) + b \sin(\beta)) \sin(\theta)\right] + ab (\cos(\alpha) \cos(\beta) + \sin(\alpha) \sin(\beta)) = 0 \]

### Step 5: Use the Identity \( \cos^2(\theta) + \sin^2(\theta) = 1 \)
This simplifies to:
\[ r^2 - r \left[(a \cos(\alpha) + b \cos(\beta)) \cos(\theta) + (a \sin(\alpha) + b \sin(\beta)) \sin(\theta)\right] + ab \cos(\alpha - \beta) = 0 \]

### Step 6: Rearranging the Equation
Rearranging gives us the final polar equation of the circle:
\[ r^2 - r \left[a \cos(\alpha - \theta) + b \cos(\beta - \theta)\right] + ab \cos(\alpha - \beta) = 0 \]