Question

If the center and radius of the circle $\left|\frac{z-2}{z-3}\right|=2$ are respectively $(\alpha, \beta)$ and $\gamma$, then $3(\alpha+\beta+\gamma)$ is equal to

• A. 12
• B. 11
• C. 10
• D. 9

Answer 1

The Correct Option is A

To solve the problem, we need to analyze the given expression for the circle: \(\left|\frac{z-2}{z-3}\right|=2\). This expression represents a circle on the complex plane.

1. First, interpret the given expression for the circle: \(\left|\frac{z-2}{z-3}\right| = 2\). In complex analysis, this can be seen as a transformation of the form of a circle. For \(\left|\frac{z-a}{z-b}\right| = k\) where \(k \neq 1\), it represents a circle.
2. The expression \(\left|\frac{z-2}{z-3}\right| = 2\) implies the points \(z\) that lie on a circle with center \((x_1, y_1)\) (say).
3. Circle properties: For the expression \(\left|\frac{z-a}{z-b}\right| = k\), the center \(\frac{k^2 \cdot b - a}{k^2 - 1}\) and the radius is \(\frac{k \cdot |a-b|}{|k^2 - 1|}\).
4. Here, \(a = 2\), \(b = 3\), and \(k = 2\). - Calculate the center: \(\frac{2^2 \cdot 3 - 2}{2^2 - 1} = \frac{12 - 2}{3} = \frac{10}{3}\) - Thus, center \((\alpha, \beta) = \left(\frac{10}{3}, 0\right)\).
5. Calculate the radius: \(\frac{2 \cdot |2 - 3|}{|2^2 - 1|} = \frac{2 \cdot 1}{3} = \frac{2}{3}\). Therefore, the radius \((\gamma) = \frac{2}{3}\).
6. Thus, \(\alpha + \beta + \gamma = \frac{10}{3} + 0 + \frac{2}{3} = 4\).
7. Therefore, \(3(\alpha + \beta + \gamma) = 3 \times 4 = 12\).

Hence, the value of \(3(\alpha + \beta + \gamma)\) is 12. Thus, the correct answer is 12.