Why does $\arg(z_1 - z_2) = \frac{\pi}{4}$ imply the slope is 1?

Think of the vector $z_1 - z_2$ as $(x_1 - x_2) + i(y_1 - y_2)$. The argument is the angle $\theta$ such that $\tan(\theta) = \frac{\text{Imaginary Part}}{\text{Real Part}}$. Here, $\tan(\frac{\pi}{4}) = \frac{y_1 - y_2}{x_1 - x_2} = 1$. It is just the definition of slope in the complex plane!

JEE Main 2021 (27 Aug Shift 2)

LEVELJEE Advanced

Animated Solution for Mathematics - Complex Numbers: Let $z_{1}$ and $z_{2}$ be two complex numbers such that $ar g (z_{1} - z_{2}) = \frac{π}{4}$ and $z_{1}, z_{2}$ satisfy the equation $∣ z - 3∣ = Re (z)$ . Then the imaginary part of $z_{1} + z_{2}$ is equal to .

Q: Why can we divide by $(y_1 - y_2)$? What if $y_1 = y_2$?

Excellent attention to detail! If $y_1 = y_2$, then $x_1 = x_2$ because the slope is 1. That would mean $z_1 = z_2$. However, the problem implies two distinct points forming a line with a specific angle. If they were the same point, the argument would be undefined. So, we are safe to divide!

Enter Numerical Value:

Progressive Hints

Hint 1

The equation

∣ z - 3∣ = Re (z)

is not just an algebraic expression; it is a geometric definition. If you substitute

z = x + i y

, you will find that the distance from a point

(x, y)

(3, 0)

is equal to the distance to the vertical line

x = 0

. What conic section does this describe?

Hint 2

The condition

ar g (z_{1} - z_{2}) = \frac{π}{4}

is a powerful geometric constraint. In the Cartesian plane, the argument of the difference of two complex numbers represents the angle of the vector connecting them. This implies the slope of the line segment joining

z_{1}

and

z_{2}

tan (\frac{π}{4}) = 1

Hint 3

Since both

z_{1}

and

z_{2}

lie on the parabola

y^{2} = 6 (x - \frac{3}{2})

, write the equation for both points. Subtracting these two equations will create a term

(y_{1}^{2} - y_{2}^{2})

on one side and

6 (x_{1} - x_{2})

on the other. Use the slope condition

y_{1} - y_{2} = x_{1} - x_{2}

to simplify the expression.

Visualized Solution

Analyze the Locus Equation

Given equation: $∣ z - 3∣ = Re (z)$
This represents a locus in the complex plane.
We need to find the Cartesian form by substituting $z = x + i y$ .

Substitute $z = x + i y$

Substitute $z = x + i y$ into the equation.
LHS: $∣ (x - 3) + i y ∣$
RHS: $x$
Equation: $(x - 3)^{2} + y^{2} = x$

Square Both Sides

Square both sides to eliminate the radical:
$(x - 3)^{2} + y^{2} = x^{2}$
Expand the term $(x - 3)^{2}$ :
$x^{2} - 6 x + 9 + y^{2} = x^{2}$

Simplify to Parabola Form

Cancel $x^{2}$ from both sides:
$- 6 x + 9 + y^{2} = 0$
Rearrange to isolate $y^{2}$ :
$y^{2} = 6 x - 9$
Standard form: $y^{2} = 6 (x - \frac{3}{2})$

Interpret the Argument Condition

Given: $ar g (z_{1} - z_{2}) = \frac{π}{4}$
Let $z_{1} = x_{1} + i y_{1}$ and $z_{2} = x_{2} + i y_{2}$ .
$ar g ((x_{1} - x_{2}) + i (y_{1} - y_{2})) = \frac{π}{4}$

Express Argument as Slope

$⟹ \frac{y _{1} - y _{2}}{x _{1} - x _{2}} = tan (\frac{π}{4}) = 1$
So, $y_{1} - y_{2} = x_{1} - x_{2}$

Points on the Parabola

$z_{1}$ lies on the parabola: $y_{1}^{2} = 6 x_{1} - 9$
$z_{2}$ lies on the parabola: $y_{2}^{2} = 6 x_{2} - 9$

Subtract the Two Equations

Subtract the second equation from the first:
$y_{1}^{2} - y_{2}^{2} = (6 x_{1} - 9) - (6 x_{2} - 9)$
Simplify the right side:
$y_{1}^{2} - y_{2}^{2} = 6 (x_{1} - x_{2})$

Factorize and Substitute

Factor the LHS: $(y_{1} - y_{2}) (y_{1} + y_{2}) = 6 (x_{1} - x_{2})$
From the argument condition: $x_{1} - x_{2} = y_{1} - y_{2}$
Substitute this into the equation:
$(y_{1} - y_{2}) (y_{1} + y_{2}) = 6 (y_{1} - y_{2})$

Final Calculation

Divide both sides by $(y_{1} - y_{2})$ , assuming $z_{1} \neq = z_{2}$ :
$y_{1} + y_{2} = 6$
The imaginary part of $z_{1} + z_{2}$ is $Im (z_{1} + z_{2}) = y_{1} + y_{2}$ .
Final Answer: $6$

00:00 / 00:00

The Sigma Insight: Geometrical Applications of Complex Numbers

Solution Diagram

Analyzing the Setup

We begin by interpreting the equation

∣ z - 3∣ = Re (z)

within the complex plane. By substituting

z = x + i y

, the equation transforms into the distance formula:

(x - 3)^{2} + y^{2} = x

Squaring both sides of the equation yields:

(x - 3)^{2} + y^{2} = x^{2}

Expanding the binomial term, we obtain

x^{2} - 6 x + 9 + y^{2} = x^{2}

. The

x^{2}

terms cancel out, simplifying the expression to:

y^{2} = 6 x - 9 \Rightarrow y^{2} = 6 (x - \frac{3}{2})

This confirms that the locus of points

z

is a parabola opening to the right.

The Argument as a Geometric Bridge

Next, we consider the condition

ar g (z_{1} - z_{2}) = \frac{π}{4}

. This geometric constraint dictates that the slope of the line segment connecting

z_{1}

and

z_{2}

is:

tan (\frac{π}{4}) = 1

Given

z_{1} = x_{1} + i y_{1}

and

z_{2} = x_{2} + i y_{2}

, the slope condition implies:

\frac{y _{1} - y _{2}}{x _{1} - x _{2}} = 1 \Rightarrow y_{1} - y_{2} = x_{1} - x_{2}

This relationship serves as the critical bridge between the complex coordinates and the linear geometry of the plane.

The Algebraic Symphony

Since both

z_{1}

and

z_{2}

lie on the parabola, they must satisfy the equation

y^{2} = 6 x - 9

. We write this for both points:

y_{1}^{2} = 6 x_{1} - 9

y_{2}^{2} = 6 x_{2} - 9

Subtracting these two equations eliminates the constant term:

y_{1}^{2} - y_{2}^{2} = 6 (x_{1} - x_{2})

Factoring the left side as

(y_{1} - y_{2}) (y_{1} + y_{2})

and substituting the slope condition

x_{1} - x_{2} = y_{1} - y_{2}

, we get:

(y_{1} - y_{2}) (y_{1} + y_{2}) = 6 (y_{1} - y_{2})

Assuming $z_1 eq z_2$, we divide by the non-zero difference

(y_{1} - y_{2})

to arrive at the final result:

y_{1} + y_{2} = 6

The imaginary part of the sum

z_{1} + z_{2}

is defined as

y_{1} + y_{2}

. Therefore, the value is 6.

Common Traps & FAQs

Wait, why is

∣ z - 3∣ = Re (z)

a parabola? I thought it was a circle!

That is a fantastic observation! A circle is defined by a constant distance to a fixed point, like

∣ z - 3∣ = R

. Here, the right-hand side is not a constant; it is the real part of

z

, which is

x

. When the distance to a fixed point equals the distance to a fixed line (in this case, the y-axis,

x = 0

), that is the classic definition of a parabola. Keep visualizing!

Why can we divide by

(y_{1} - y_{2})

? What if

y_{1} = y_{2}

Excellent attention to detail! If

y_{1} = y_{2}

, then

x_{1} = x_{2}

because the slope is 1. That would mean

z_{1} = z_{2}

. However, the problem implies two distinct points forming a line with a specific angle. If they were the same point, the argument would be undefined. So, we are safe to divide!

Why does

ar g (z_{1} - z_{2}) = \frac{π}{4}

imply the slope is 1?

Think of the vector

z_{1} - z_{2}

(x_{1} - x_{2}) + i (y_{1} - y_{2})

. The argument is the angle

θ

such that

tan (θ) = \frac{Imaginary Part}{Real Part}

. Here,

tan (\frac{π}{4}) = \frac{y _{1} - y _{2}}{x _{1} - x _{2}} = 1

. It is just the definition of slope in the complex plane!

Animated Solution for Mathematics - Complex Numbers: Let $z_{1}$ and $z_{2}$ be two complex numbers such that $ar g (z_{1} - z_{2}) = \frac{π}{4}$ and $z_{1}, z_{2}$ satisfy the equation $∣ z - 3∣ = Re (z)$ . Then the imaginary part of $z_{1} + z_{2}$ is equal to .

Enter Numerical Value:

Visualized Solution

Analyze the Locus Equation

Substitute $z = x + i y$

Square Both Sides

Simplify to Parabola Form

Interpret the Argument Condition

Express Argument as Slope

Points on the Parabola

Subtract the Two Equations

Factorize and Substitute

Final Calculation

The Sigma Insight: Geometrical Applications of Complex Numbers

Analyzing the Setup

The Argument as a Geometric Bridge

The Algebraic Symphony

Similar Questions

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If $z_{1}$ and $z_{2}$ are two non-zero complex numbers such that $∣ z_{1} + z_{2} ∣ = ∣ z_{1} ∣ + ∣ z_{2} ∣$ , then $a r g z_{1} - a r g z_{2}$ is equal to

If $z_{1}$ and $z_{2}$ are two nonzero complex numbers such that $∣ z_{1} + z_{2} ∣ = ∣ z_{1} ∣ + ∣ z_{2} ∣$ , then $A r g z_{1} - A r g z_{2}$ is equal to

Let $z_{1} = 10 + 6 i$ and $z_{2} = 4 + 6 i$ . If $z$ is any complex number such that the argument of $\frac{z - z _{1}}{z - z _{2}}$ is $\frac{π}{4}$ , then prove that $∣ z - 7 - 9 i ∣ = 32$ .

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Enter Numerical Value:

Visualized Solution

Analyze the Locus Equation

Substitute .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } z=x+iy

Square Both Sides

Simplify to Parabola Form

Interpret the Argument Condition

Express Argument as Slope

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Subtract the Two Equations

Factorize and Substitute

Final Calculation

The Sigma Insight: Geometrical Applications of Complex Numbers

Analyzing the Setup

The Argument as a Geometric Bridge

The Algebraic Symphony

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