If `|z-1-i|=1`, then the locus of a point represented by the complex number `5(z-i)-6` is

Options

A. circle with centre `(1,0)` and radius `3`
B. circle with centre `(-1,0)` and radius `5`
C. line passing through origin
D. line passing through `(-1,0)`

Explanation

To solve the problem, we need to find the locus of the point represented by the complex number \(5(z - i) - 6\) given that \(|z - 1 - i| = 1\).

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
The equation \(|z - 1 - i| = 1\) describes a circle in the complex plane. The center of this circle is at the point \(1 + i\) (which corresponds to the complex number \(1 + i\)) and the radius is \(1\).

2. **Expressing \(z\)**:
Let \(z = x + iy\), where \(x\) and \(y\) are real numbers. The condition can be rewritten as:
\[ |(x + iy) - (1 + i)| = 1 \]
This simplifies to:
\[ |(x - 1) + i(y - 1)| = 1 \]
The modulus gives us:
\[ \sqrt{(x - 1)^2 + (y - 1)^2} = 1 \]
Squaring both sides, we have:
\[ (x - 1)^2 + (y - 1)^2 = 1 \]
This is the equation of a circle centered at \((1, 1)\) with radius \(1\).

3. **Finding the Locus of \(5(z - i) - 6\)**:
Now, we need to find the expression for \(5(z - i) - 6\):
\[ 5(z - i) - 6 = 5((x + iy) - i) - 6 = 5(x + (y - 1)i) - 6 \]
This simplifies to:
\[ 5x + 5(y - 1)i - 6 = (5x - 6) + 5(y - 1)i \]
Let’s denote this new complex number as \(\omega\):
\[ \omega = (5x - 6) + 5(y - 1)i \]

4. **Finding the Locus in Terms of \(\omega\)**:
We can express \(\omega\) in terms of its real and imaginary parts:
- Real part: \(u = 5x - 6\)
- Imaginary part: \(v = 5(y - 1)\)

From the equation of the circle \((x - 1)^2 + (y - 1)^2 = 1\), we can express \(y\) in terms of \(x\):
\[ y = 1 \pm \sqrt{1 - (x - 1)^2} \]

5. **Substituting \(y\) into the Locus**:
Substitute \(y\) back into the expression for \(v\):
\[ v = 5\left(1 \pm \sqrt{1 - (x - 1)^2} - 1\right) = 5\left(\pm \sqrt{1 - (x - 1)^2}\right) \]
Thus, \(v\) can take values from \(-5\) to \(5\).

6. **Finding the Center and Radius**:
The locus of \(\omega\) can be expressed as:
\[ (u + 6)^2 + \left(\frac{v}{5}\right)^2 = 1 \]
This represents a circle centered at \((-6, 0)\) with a radius of \(5\).

### Final Answer:
The locus of the point represented by the complex number \(5(z - i) - 6\) is a circle with center \((-6, 0)\) and radius \(5\).