If `z_(1)` lies on |z| = r, then equation of tangent at `z_(1)` is

Options

A. `(z)/(bar(z)_(1)) + (bar(z))/(bar(z)_(1)) = 2`
B. `(z)/(bar(z)_(1)) + (bar(z))/(bar(z)_(1)) = r`
C. `(z)/(bar(z)_(1)) + (bar(z))/(z_(1)) = 2`
D. `(z)/(bar(z)_(1)) + (bar(z))/(z_(1)) = r`

Answer

To find the equation of the tangent at the point \( z_1 \) on the circle defined by \( |z| = r \), we can follow these steps:

### Step 1: Understand the Circle
The equation \( |z| = r \) represents a circle centered at the origin (0, 0) with a radius \( r \). Any point \( z_1 \) on this circle can be expressed in terms of its coordinates as:
\[ z_1 = x_1 + i y_1 \]
where \( x_1^2 + y_1^2 = r^2 \).

### Step 2: Equation of the Tangent
The equation of the tangent to a circle at a point \( z_1 \) can be derived using the formula:
\[ z \cdot \overline{z_1} = |z_1|^2 \]
Given that \( |z_1|^2 = r^2 \) (since \( z_1 \) lies on the circle), we can write:
\[ z \cdot \overline{z_1} = r^2 \]

### Step 3: Substitute \( z_1 \)
Substituting \( z_1 = x_1 + i y_1 \) into the equation gives:
\[ z \cdot (x_1 - i y_1) = r^2 \]
This can be expanded as:
\[ z \cdot x_1 - z \cdot i y_1 = r^2 \]

### Step 4: Express in Terms of \( z \)
The equation can be rearranged to express \( z \):
\[ z \cdot x_1 - i z \cdot y_1 = r^2 \]
This shows that the tangent line at point \( z_1 \) can be represented in terms of \( z \) and \( \overline{z_1} \).

### Step 5: Final Form
Thus, the final equation of the tangent at point \( z_1 \) on the circle \( |z| = r \) can be expressed as:
\[ z \cdot \overline{z_1} = r^2 \]
or equivalently:
\[ z \cdot z_1^* = r^2 \]
where \( z_1^* \) is the conjugate of \( z_1 \).

### Conclusion
The equation of the tangent line at the point \( z_1 \) on the circle \( |z| = r \) is:
\[ z \cdot \overline{z_1} = r^2 \]

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