if `|z|=3` then the points representing thecomplex numbers `-1+4z` lie on a

Options

A. line
B. circle
C. parabola
D. none of these

Explanation

To solve the problem, we need to determine the geometric representation of the complex numbers of the form \(-1 + 4z\) given that \(|z| = 3\).

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We know that \(|z| = 3\). This means that the complex number \(z\) lies on a circle of radius 3 centered at the origin in the complex plane.

2. **Expressing the Complex Number**:
We need to analyze the expression \(-1 + 4z\). Let's denote this complex number as \(a\):
\[ a = -1 + 4z \]

3. **Finding the Modulus**:
We can find the modulus of \(4z\):
\[ |4z| = 4|z| = 4 \times 3 = 12 \]
Thus, the modulus of \(4z\) is 12.

4. **Relating the Modulus to \(a\)**:
We can express the modulus of \(a\):
\[ |a + 1| = |4z| \]
This means:
\[ |a + 1| = 12 \]

5. **Setting Up the Equation**:
Let \(a = x + iy\) where \(x\) and \(y\) are real numbers. Then:
\[ |(x + 1) + iy| = 12 \]
This can be rewritten as:
\[ \sqrt{(x + 1)^2 + y^2} = 12 \]

6. **Squaring Both Sides**:
Squaring both sides gives:
\[ (x + 1)^2 + y^2 = 144 \]

7. **Identifying the Geometric Shape**:
The equation \((x + 1)^2 + y^2 = 144\) represents a circle in the complex plane. The center of this circle is at \((-1, 0)\) and the radius is \(12\).

### Conclusion:
The points representing the complex numbers \(-1 + 4z\) lie on a circle with center at \((-1, 0)\) and radius \(12\).