For all complex numbers z of the form 1 + i$\alpha$, $\alpha$ $\epsilon$ R, if z = x + iy, then :

Question

If $50 \left( \frac{2x}{1 + 3i} + \frac{y}{1 - 2i} \right) = 31 + 17i$ where $x, y \in R$ & $i = \sqrt{-1}$ then value of $10(x + 3y)$ is ________

Answer 1

To solve this problem, we need to analyze the given information and derive the equation that satisfies the complex number condition.

We have complex numbers of the form $ z = 1 + i\alpha $, where $ \alpha \in \mathbb{R} $.
The complex number is expressed in the form $ z = x + iy $, where $ x $ and $ y $ are real numbers.
By comparing the two forms of $ z $, we have $ x = 1 $ and $ y = \alpha $.
We need to find a relationship involving $ x $ and $ y $ that matches one of the given options.
The problem provides four potential relationships. To identify the correct one, we substitute $ x = 1 $ into each option and simplify:

Option 1: $ y^2 - 4x + 2 = 0 $
- Substituting: $ y^2 - 4(1) + 2 = y^2 - 4 + 2 = y^2 - 2 = 0 $
- This simplifies to $ y^2 = 2 $. This is not a general relationship involving $ y $.
Option 2: $ y^2 + 4x - 4 = 0 $
- Substituting: $ y^2 + 4(1) - 4 = y^2 + 4 - 4 = y^2 = 0 $
- This simplifies correctly since for $ y = \alpha $, which implies $ \alpha^2 = 0 $, is universally true because $ \alpha $ can be zero.
Option 3: $ y^2 - 4x + 4 = 0 $
- Substituting: $ y^2 - 4(1) + 4 = y^2 - 4 + 4 = y^2 = 0 $
- This condition is too strict, requiring $ y = 0 $, which is not applicable generally.
Option 4: $ y^2 + 4x + 2 = 0 $
- Substituting: $ y^2 + 4(1) + 2 = y^2 + 4 + 2 = y^2 + 6 = 0 $
- This condition requires $ y^2 = -6 $, which is not possible for real $ y $.

The only equation that accounts for the basic composition of the number in general without imposing impossible conditions is:

The correct answer is $ y^2 + 4x - 4 = 0 $.

Answer 2

To solve this problem, we need to analyze the given information and derive the equation that satisfies the complex number condition.

We have complex numbers of the form $ z = 1 + i\alpha $, where $ \alpha \in \mathbb{R} $.
The complex number is expressed in the form $ z = x + iy $, where $ x $ and $ y $ are real numbers.
By comparing the two forms of $ z $, we have $ x = 1 $ and $ y = \alpha $.
We need to find a relationship involving $ x $ and $ y $ that matches one of the given options.
The problem provides four potential relationships. To identify the correct one, we substitute $ x = 1 $ into each option and simplify:

Option 1: $ y^2 - 4x + 2 = 0 $
- Substituting: $ y^2 - 4(1) + 2 = y^2 - 4 + 2 = y^2 - 2 = 0 $
- This simplifies to $ y^2 = 2 $. This is not a general relationship involving $ y $.
Option 2: $ y^2 + 4x - 4 = 0 $
- Substituting: $ y^2 + 4(1) - 4 = y^2 + 4 - 4 = y^2 = 0 $
- This simplifies correctly since for $ y = \alpha $, which implies $ \alpha^2 = 0 $, is universally true because $ \alpha $ can be zero.
Option 3: $ y^2 - 4x + 4 = 0 $
- Substituting: $ y^2 - 4(1) + 4 = y^2 - 4 + 4 = y^2 = 0 $
- This condition is too strict, requiring $ y = 0 $, which is not applicable generally.
Option 4: $ y^2 + 4x + 2 = 0 $
- Substituting: $ y^2 + 4(1) + 2 = y^2 + 4 + 2 = y^2 + 6 = 0 $
- This condition requires $ y^2 = -6 $, which is not possible for real $ y $.

The only equation that accounts for the basic composition of the number in general without imposing impossible conditions is:

The correct answer is $ y^2 + 4x - 4 = 0 $.

The Correct Option is B