If the roots of the equation
`x^4 - 10x^3 +50 x^2 - 130 x + 169 = 0` are of the form ` a +- ib ` and ` b +- ia ` then (a,b)=

Options

A. `(3,2)`
B. `(2,1)`
C. `(-3,2)`
D. `(-3,-2)`

Answer

To solve the equation \( x^4 - 10x^3 + 50x^2 - 130x + 169 = 0 \) given that the roots are of the form \( a \pm ib \) and \( b \pm ia \), we can follow these steps:

### Step 1: Identify the form of the roots
The roots are given as \( a + ib \) and \( a - ib \) (which we can denote as \( r_1 \) and \( r_2 \)), and \( b + ia \) and \( b - ia \) (denote these as \( r_3 \) and \( r_4 \)).

### Step 2: Use Vieta's formulas
According to Vieta's formulas, the sum of the roots (which is \( r_1 + r_2 + r_3 + r_4 \)) is equal to the coefficient of \( x^3 \) with a negative sign. Thus:

\[ r_1 + r_2 + r_3 + r_4 = 10 \]

Since \( r_1 + r_2 = 2a \) and \( r_3 + r_4 = 2b \), we have:

\[ 2a + 2b = 10 \]

### Step 3: Simplify the equation
Dividing the entire equation by 2 gives:

\[ a + b = 5 \]

### Step 4: Find possible values for \( a \) and \( b \)
Now we need to find pairs \( (a, b) \) such that \( a + b = 5 \). We can check the options provided in the question.

### Step 5: Check options
Let's check the possible pairs:
1. If \( a = 3 \) and \( b = 2 \), then \( a + b = 3 + 2 = 5 \) (valid).
2. If \( a = 4 \) and \( b = 1 \), then \( a + b = 4 + 1 = 5 \) (valid).
3. If \( a = 5 \) and \( b = 0 \), then \( a + b = 5 + 0 = 5 \) (valid).
4. If \( a = 2 \) and \( b = 3 \), then \( a + b = 2 + 3 = 5 \) (valid).

However, the question specifies that we need to find a unique pair \( (a, b) \) that satisfies the conditions of the roots.

### Step 6: Conclude the values
From the options provided, the pair \( (3, 2) \) is the only one that matches the requirement of the problem statement.

Thus, the values of \( (a, b) \) are:

\[ (a, b) = (3, 2) \]