Let P(h, k) be a point on an argand plane equidistant from the roots of the equation `(z+1)^(4)=16z^(4)`, then the value of h is equal to

Options

A. 0
B. `(2)/(3)`
C. `(1)/(3)`
D. `(1)/(2)`

Answer

To solve the problem, we need to find the point \( P(h, k) \) on the Argand plane that is equidistant from the roots of the equation \( (z + 1)^4 = 16z^4 \).

### Step 1: Rewrite the equation
We start with the equation:
\[ (z + 1)^4 = 16z^4 \]
Taking the fourth root on both sides, we can express this as:
\[ z + 1 = 2z \quad \text{or} \quad z + 1 = -2z \quad \text{or} \quad z + 1 = 2iz \quad \text{or} \quad z + 1 = -2iz \]

### Step 2: Solve for \( z \)
1. **For \( z + 1 = 2z \)**:
\[ 1 = z \implies z = 1 \]

2. **For \( z + 1 = -2z \)**:
\[ z + 2z = -1 \implies 3z = -1 \implies z = -\frac{1}{3} \]

3. **For \( z + 1 = 2iz \)**:
\[ z - 2iz = -1 \implies z(1 - 2i) = -1 \implies z = \frac{-1}{1 - 2i} \]
To simplify:
\[ z = \frac{-1(1 + 2i)}{(1 - 2i)(1 + 2i)} = \frac{-1 - 2i}{1 + 4} = \frac{-1 - 2i}{5} = -\frac{1}{5} - \frac{2}{5}i \]

4. **For \( z + 1 = -2iz \)**:
\[ z + 2iz = -1 \implies z(1 + 2i) = -1 \implies z = \frac{-1}{1 + 2i} \]
To simplify:
\[ z = \frac{-1(1 - 2i)}{(1 + 2i)(1 - 2i)} = \frac{-1 + 2i}{1 + 4} = \frac{-1 + 2i}{5} = -\frac{1}{5} + \frac{2}{5}i \]

### Step 3: List the roots
The roots of the equation are:
1. \( z = 1 \)
2. \( z = -\frac{1}{3} \)
3. \( z = -\frac{1}{5} - \frac{2}{5}i \)
4. \( z = -\frac{1}{5} + \frac{2}{5}i \)

### Step 4: Find the midpoint of the real parts
The real parts of the roots are:
- \( 1 \)
- \( -\frac{1}{3} \)
- \( -\frac{1}{5} \)
- \( -\frac{1}{5} \)

To find the midpoint of the real parts:
\[ \text{Midpoint} = \frac{1 + \left(-\frac{1}{3}\right) + \left(-\frac{1}{5}\right) + \left(-\frac{1}{5}\right)}{4} \]

Calculating:
\[ = \frac{1 - \frac{1}{3} - \frac{1}{5} - \frac{1}{5}}{4} \]
Finding a common denominator (15):
\[ = \frac{\frac{15}{15} - \frac{5}{15} - \frac{3}{15} - \frac{3}{15}}{4} = \frac{\frac{15 - 5 - 3 - 3}{15}}{4} = \frac{\frac{4}{15}}{4} = \frac{1}{15} \]

### Step 5: Conclusion
The value of \( h \) is the real part of the midpoint, which is:
\[ h = \frac{1}{15} \]