Let z1 and z2 be two distinct complex numbers and let z = (1-t)z1 + tz2 for some real number t with

Q: Wait, why is $Arg(z - z_1) = Arg(z - z_2)$ incorrect? They are on the same line, right?

You are absolutely right that they are collinear! However, collinearity does not mean the vectors point in the same direction. The vector $(z - z_1)$ points from $z_1$ to $z$, while $(z - z_2)$ points from $z_2$ to $z$. Since $z$ is between $z_1$ and $z_2$, these vectors point in opposite directions. Therefore, their arguments differ by $\pi$ radians, not zero.

Q: How do I know if the determinant form in Option (C) is the correct way to represent collinearity?

Great question! The condition for three points $z, z_1, z_2$ to be collinear is that the slope of the segment $zz_1$ must equal the slope of $z_1z_2$. In complex numbers, this is expressed as $\frac{z - z_1}{z_2 - z_1} = \frac{\bar{z} - \bar{z}_1}{\bar{z}_2 - \bar{z}_1}$. If you cross-multiply this and rearrange the terms, you get $(z - z_1)(\bar{z}_2 - \bar{z}_1) - (z_2 - z_1)(\bar{z} - \bar{z}_1) = 0$, which is exactly the expansion of the determinant given in Option (C).

Q: Can I just pick values like $z_1 = 0$ and $z_2 = 1$ to test the options?

Yes, you can! This is a fantastic strategy for competitive exams. If you set $z_1 = 0$ and $z_2 = 1$, then $z = t$. You can then check each option. For example, $|t - 0| + |t - 1| = t + (1 - t) = 1$, which is $|z_1 - z_2|$. This confirms Option (A). While this is a great way to verify, understanding the geometric 'why' is what will help you solve more complex, non-testable problems in the future.

Question

Let $z_1$ and $z_2$ be two distinct complex numbers and let $z = (1-t)z_1 + tz_2$ for some real number $t$ with $0 < t < 1$.

Options:

(A) $|z - z_1| + |z - z_2| = |z_1 - z_2|$

(B) $Arg(z - z_1) = Arg(z - z_2)$

(C) $\begin{vmatrix} z - z_1 & \bar{z} - \bar{z}_1 \ z_2 - z_1 & \bar{z}_2 - \bar{z}_1 \end{vmatrix} = 0$

(D) $Arg(z - z_1) = Arg(z_2 - z_1)$

Sigma Percentile · Accepted Answer

### Analyzing the Geometry of Complex Numbers

Welcome, fellow traveler on the path to JEE mastery. Today, we are not just solving an algebraic equation; we are uncovering the hidden geometric soul of complex numbers.

When you see an expression like $z = (1-t)z_1 + tz_2$, I want you to stop seeing variables and start seeing a physical reality. Imagine you are standing on a vast, flat plane with two fixed markers, $z_1$ and $z_2$.

The equation $z = (1-t)z_1 + tz_2$ is a recipe for finding a point $z$ that sits perfectly on the straight line connecting those two markers. Because $0 < t < 1$, this point $z$ is trapped, beautifully and precisely, between $z_1$ and $z_2$. This is the section formula in its most elegant, complex form.

### The Distance Property: Walking the Line

Let us look at Option (A): $|z - z_1| + |z - z_2| = |z_1 - z_2|$. In the language of complex numbers, the modulus $|z_a - z_b|$ is simply the distance between two points.

This equation states that the distance from $z_1$ to $z$ plus the distance from $z$ to $z_2$ equals the total distance from $z_1$ to $z_2$. If you are walking from $z_1$ to $z_2$ and you stop at $z$ along the way, your total distance traveled is exactly the length of the path.

This confirms that $z$ must lie on the segment $z_1z_2$. It is a beautiful, intuitive truth that holds firm in the complex plane.

### The Vector Argument: Direction Matters

Now, let us tackle the argument, or the angle, of these complex numbers. If we rearrange our original equation, we get $z - z_1 = t(z_2 - z_1)$.

This is a powerful vector relation. Since $t$ is a positive real number, multiplying the vector $(z_2 - z_1)$ by $t$ changes its length but leaves its direction untouched. This means the vector $(z - z_1)$ points in the exact same direction as $(z_2 - z_1)$.

Consequently, their arguments must be identical: $Arg(z - z_1) = Arg(z_2 - z_1)$. This validates Option (D).

However, be careful! If you look at Option (B), it asks if $Arg(z - z_1) = Arg(z - z_2)$. We just established that $(z - z_1)$ points from $z_1$ toward $z$, but $(z - z_2)$ points from $z_2$ toward $z$.

Since $z$ is between them, these two vectors are pointing in opposite directions. They are like two people standing on a line, facing away from each other. Their arguments differ by $\pi$ radians, not zero. Thus, Option (B) is a classic trap—do not fall for it!

### The Elegance of Collinearity

Finally, we arrive at the determinant in Option (C). We have already established that $z, z_1,$ and $z_2$ are collinear—they all live on the same line.

In the complex plane, the condition for three points to be collinear is that the ratio of the differences must be purely real. That is:

$$\frac{z - z_1}{z_2 - z_1} = \frac{\bar{z} - \bar{z}_1}{\bar{z}_2 - \bar{z}_1}$$

If you take this equation and cross-multiply, you get:

$$(z - z_1)(\bar{z}_2 - \bar{z}_1) = (z_2 - z_1)(\bar{z} - \bar{z}_1)$$

Moving everything to one side, we get:

$$(z - z_1)(\bar{z}_2 - \bar{z}_1) - (z_2 - z_1)(\bar{z} - \bar{z}_1) = 0$$

This is the expansion of the determinant:

$$\begin{vmatrix} z - z_1 & \bar{z} - \bar{z}_1 \ z_2 - z_1 & \bar{z}_2 - \bar{z}_1 \end{vmatrix} = 0$$

It is a perfect, symmetrical conclusion. We have used geometry, vectors, and algebra to prove that $z$ is not just a point, but a part of a beautiful, linear relationship. Keep this geometric intuition in your toolkit, and no complex number problem will ever be able to hide its secrets from you again.

Animated Solution for Mathematics - Complex Numbers: Let $z_{1}$ and $z_{2}$ be two distinct complex numbers and let $z = (1 - t) z_{1} + t z_{2}$ for some real number $t$ with $0 < t < 1$ .

Select Answer:

Visualized Solution

Visualizing the Complex Plane

The Given Equation

The Section Formula

Internal Division

Analyzing Option A: Distances

Verifying Option A

Analyzing Option D: Vectors

Verifying Option D

Analyzing Option B

Verifying Option B

Analyzing Option C: Collinearity

The Collinearity Determinant

Verifying Option C

The Sigma Insight: Geometrical Applications of Complex Numbers

Analyzing the Geometry of Complex Numbers

The Distance Property

The Vector Argument

The Elegance of Collinearity

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Animated Solution for Mathematics - Complex Numbers: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z1​ and z2​ be two distinct complex numbers and let z=(1−t)z1​+tz2​ for some real number t with 0<t<1.