Question:
If $ z_1 = 2 + 3i $, $ z_2 = 4 - 5i $, and $ z_3 $ are three points in the Argand plane such that $ 5z_1 + xz_2 + yz_3 = 0 $ (where $ x, y \in \mathbb{R} $) and $ z_3 $ is the midpoint of the line segment joining the points $ z_1 $ and $ z_2 $, then find $ x + y $.
If $ z_1 = 2 + 3i $, $ z_2 = 4 - 5i $, and $ z_3 $ are three points in the Argand plane such that $ 5z_1 + xz_2 + yz_3 = 0 $ (where $ x, y \in \mathbb{R} $) and $ z_3 $ is the midpoint of the line segment joining the points $ z_1 $ and $ z_2 $, then find $ x + y $.
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When working with complex numbers in the Argand plane, always remember that the real and imaginary parts must be treated separately. Solve for each part (real and imaginary) and then combine the results.
Updated On: May 9, 2025
- \( -5 \)
- \( 0 \)
- \( 4 \)
- \( -1 \)
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The Correct Option is A
Solution and Explanation
Step 1: Use the condition that \( z_3 \) is the midpoint of \( z_1 \) and \( z_2 \).
The midpoint formula for complex numbers is: \[ z_3 = \frac{z_1 + z_2}{2} \] Substitute the values of \( z_1 \) and \( z_2 \): \[ z_3 = \frac{(2 + 3i) + (4 - 5i)}{2} = \frac{6 - 2i}{2} = 3 - i \] So, \( z_3 = 3 - i \).
Step 2: Substitute \( z_3 \) into the given equation \( 5z_1 + xz_2 + yz_3 = 0 \).
Substitute \( z_1 = 2 + 3i \), \( z_2 = 4 - 5i \), and \( z_3 = 3 - i \) into the equation: \[ 5(2 + 3i) + x(4 - 5i) + y(3 - i) = 0 \] This simplifies to: \[ (10 + 15i) + x(4 - 5i) + y(3 - i) = 0 \] Expanding the terms: \[ 10 + 15i + 4x - 5xi + 3y - yi = 0 \] Combine real and imaginary parts: \[ (10 + 4x + 3y) + (15 - 5x - y)i = 0 \]
Step 3: Solve for \( x \) and \( y \).
For the equation to be true, both the real and imaginary parts must be zero. This gives us the system of equations: \[ 10 + 4x + 3y = 0 \quad \text{(real part)} \] \[ 15 - 5x - y = 0 \quad \text{(imaginary part)} \]
Step 4: Solve the system of equations.
From the second equation, solve for \( y \): \[ y = 15 - 5x \] Substitute this into the first equation: \[ 10 + 4x + 3(15 - 5x) = 0 \] Simplify: \[ 10 + 4x + 45 - 15x = 0 \] \[ 55 - 11x = 0 \] \[ x = 5 \] Now substitute \( x = 5 \) into \( y = 15 - 5x \): \[ y = 15 - 5(5) = 15 - 25 = -10 \]
Step 5: Calculate \( x + y \). \[ x + y = 5 + (-10) = -5 \] Thus, the value of \( x + y \) is \( \boxed{-5} \).
The midpoint formula for complex numbers is: \[ z_3 = \frac{z_1 + z_2}{2} \] Substitute the values of \( z_1 \) and \( z_2 \): \[ z_3 = \frac{(2 + 3i) + (4 - 5i)}{2} = \frac{6 - 2i}{2} = 3 - i \] So, \( z_3 = 3 - i \).
Step 2: Substitute \( z_3 \) into the given equation \( 5z_1 + xz_2 + yz_3 = 0 \).
Substitute \( z_1 = 2 + 3i \), \( z_2 = 4 - 5i \), and \( z_3 = 3 - i \) into the equation: \[ 5(2 + 3i) + x(4 - 5i) + y(3 - i) = 0 \] This simplifies to: \[ (10 + 15i) + x(4 - 5i) + y(3 - i) = 0 \] Expanding the terms: \[ 10 + 15i + 4x - 5xi + 3y - yi = 0 \] Combine real and imaginary parts: \[ (10 + 4x + 3y) + (15 - 5x - y)i = 0 \]
Step 3: Solve for \( x \) and \( y \).
For the equation to be true, both the real and imaginary parts must be zero. This gives us the system of equations: \[ 10 + 4x + 3y = 0 \quad \text{(real part)} \] \[ 15 - 5x - y = 0 \quad \text{(imaginary part)} \]
Step 4: Solve the system of equations.
From the second equation, solve for \( y \): \[ y = 15 - 5x \] Substitute this into the first equation: \[ 10 + 4x + 3(15 - 5x) = 0 \] Simplify: \[ 10 + 4x + 45 - 15x = 0 \] \[ 55 - 11x = 0 \] \[ x = 5 \] Now substitute \( x = 5 \) into \( y = 15 - 5x \): \[ y = 15 - 5(5) = 15 - 25 = -10 \]
Step 5: Calculate \( x + y \). \[ x + y = 5 + (-10) = -5 \] Thus, the value of \( x + y \) is \( \boxed{-5} \).
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