Locate the point representing the complex number z on the Argand diagram for which
`|i-1-2z| gt 9`.

Answer

To locate the point representing the complex number \( z \) on the Argand diagram for which \( |i - 1 - 2z| > 9 \), we can follow these steps:

### Step 1: Rewrite the inequality
We start with the given inequality:
\[ |i - 1 - 2z| > 9 \]

### Step 2: Rearrange the expression
We can express the inequality in a more manageable form. We rewrite it as:
\[ | -2z + (i - 1) | > 9 \]
This can be rearranged to:
\[ | -2(z - \frac{i - 1}{2}) | > 9 \]

### Step 3: Simplify the expression
Since the absolute value of a product is the product of the absolute values, we have:
\[ 2 |z - \frac{i - 1}{2}| > 9 \]
Dividing both sides by 2 gives:
\[ |z - \frac{i - 1}{2}| > \frac{9}{2} \]

### Step 4: Identify the center and radius
Now, we can identify that this represents all points \( z \) that are outside a circle in the Argand plane. The center of this circle is:
\[ \frac{i - 1}{2} = \left(-\frac{1}{2}, \frac{1}{2}\right) \]
And the radius is:
\[ \frac{9}{2} \]

### Step 5: Sketch the Argand diagram
On the Argand diagram:
- The center of the circle is at the point \((-0.5, 0.5)\).
- The radius is \(4.5\) (since \(\frac{9}{2} = 4.5\)), meaning the circle extends 4.5 units from the center in all directions.

### Step 6: Conclusion
The solution to the inequality \( |i - 1 - 2z| > 9 \) indicates that the points representing \( z \) lie outside the circle centered at \((-0.5, 0.5)\) with a radius of \(4.5\).

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