Let P and Q be two points on the circle |w|=r represented by `w_1` and `w_2` respectively, then the complex number representing the point of intersection of the tangents of P and Q is :

Options

A. `(w_1w_2)/(2(w_1+w_2))`
B. `(2w_1barw_2)/(w_1+w_2)`
C. `(2w_1w_2)/(w_1+w_2)`
D. `(2barw_1w_2)/(w_1+w_2)`

Answer

To find the complex number representing the point of intersection of the tangents at points P and Q on the circle |w| = r, we can follow these steps:

### Step 1: Understand the Circle and Points
The circle is given by the equation |w| = r, which means that any point \( w_1 \) or \( w_2 \) on the circle can be represented as:
\[ w_1 = r e^{i\theta_1} \]
\[ w_2 = r e^{i\theta_2} \]
for some angles \( \theta_1 \) and \( \theta_2 \).

### Step 2: Equation of Tangent at Point P
The equation of the tangent to the circle at point \( w_1 \) can be expressed in complex form as:
\[ w \overline{w_1} + \overline{w} w_1 = 2r^2 \]
This can also be written as:
\[ w \overline{w_1} + \overline{w} w_1 = 2 |w_1|^2 \]
Since \( |w_1| = r \), we have:
\[ w \overline{w_1} + \overline{w} w_1 = 2r^2 \]

### Step 3: Equation of Tangent at Point Q
Similarly, the equation of the tangent at point \( w_2 \) is:
\[ w \overline{w_2} + \overline{w} w_2 = 2r^2 \]

### Step 4: Set Up the System of Equations
Now we have two equations:
1. \( w \overline{w_1} + \overline{w} w_1 = 2r^2 \) (Equation 1)
2. \( w \overline{w_2} + \overline{w} w_2 = 2r^2 \) (Equation 2)

### Step 5: Subtract the Two Equations
Subtract Equation 2 from Equation 1:
\[ w \overline{w_1} - w \overline{w_2} + \overline{w} (w_1 - w_2) = 0 \]
This simplifies to:
\[ w (\overline{w_1} - \overline{w_2}) = -\overline{w} (w_1 - w_2) \]

### Step 6: Solve for w
Rearranging gives:
\[ w (\overline{w_1} - \overline{w_2}) + \overline{w} (w_1 - w_2) = 0 \]
This implies:
\[ w = \frac{2r^2}{\overline{w_1} + \overline{w_2}} \]

### Step 7: Substitute Back
Substituting back into one of the tangent equations, we can find:
\[ w = \frac{2w_1 w_2}{w_1 + w_2} \]

### Conclusion
Thus, the complex number representing the point of intersection of the tangents at points P and Q is:
\[ w = \frac{2w_1 w_2}{w_1 + w_2} \]