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More from Complex Numbers and Quadratic Equations
MathMedAdv 2010 • Paper 1

Collinear Relation and Properties of Complex Linear Combination

Let z1z_1 and z2z_2 be two distinct complex numbers and let z=(1−t)z1+tz2z = (1 - t) z_1 + t z_2 for some real number tt with 0<t<10 < t < 1. If Arg(w)\text{Arg}(w) denotes the principal argument of a nonzero complex number ww, then

Options

A

∣z−z1∣+∣z−z2∣=∣z1−z2∣|z - z_1| + |z - z_2| = |z_1 - z_2|

Correct
B

Arg (z−z1)=Arg (z−z2)\text{Arg }(z - z_1) = \text{Arg }(z - z_2)

C
∣z−z1zˉ−zˉ1z2−z1zˉ2−zˉ1∣=0\left|\begin{array}{cc} z - z_1 & \bar{z} - \bar{z}_1 \\ z_2 - z_1 & \bar{z}_2 - \bar{z}_1 \end{array}\right| = 0
Correct
D

Arg (z−z1)=Arg(z2−z1)\text{Arg }(z - z_1) = \text{Arg}(z_2 - z_1)

Correct

Topics & Concepts

Complex Numbers and Quadratic EquationsComplex NumbersArgand Plane

Step-by-Step Solution

To determine which of the given statements are correct, let us analyze each option using the given condition:
z=(1−t)z1+tz2for 0<t<1z = (1 - t) z_1 + t z_2 \quad \text{for } 0 < t < 1
where z1z_1 and z2z_2 are distinct complex numbers (z1≠z2z_1 \neq z_2).

From the given expression, we can express the relative vectors:
z−z1=(1−t)z1+tz2−z1=t(z2−z1)z - z_1 = (1 - t) z_1 + t z_2 - z_1 = t (z_2 - z_1)
z−z2=(1−t)z1+tz2−z2=(1−t)(z1−z2)=−(1−t)(z2−z1)z - z_2 = (1 - t) z_1 + t z_2 - z_2 = (1 - t)(z_1 - z_2) = -(1 - t)(z_2 - z_1)

Since 0<t<10 < t < 1, both tt and 1−t1 - t are strictly positive real numbers (t>0t > 0 and 1−t>01 - t > 0). Geometrically, this means that the point representing zz lies strictly on the line segment joining the points representing z1z_1 and z2z_2 in the complex plane.

Analysis of Option A:

Taking the modulus of z−z1z - z_1 and z−z2z - z_2:
∣z−z1∣=∣t(z2−z1)∣=∣t∣⋅∣z2−z1∣=t∣z1−z2∣|z - z_1| = |t (z_2 - z_1)| = |t| \cdot |z_2 - z_1| = t |z_1 - z_2|
∣z−z2∣=∣−(1−t)(z2−z1)∣=∣1−t∣⋅∣z2−z1∣=(1−t)∣z1−z2∣|z - z_2| = |-(1 - t)(z_2 - z_1)| = |1 - t| \cdot |z_2 - z_1| = (1 - t) |z_1 - z_2|

Adding the two equations:
∣z−z1∣+∣z−z2∣=t∣z1−z2∣+(1−t)∣z1−z2∣=(t+1−t)∣z1−z2∣=∣z1−z2∣|z - z_1| + |z - z_2| = t |z_1 - z_2| + (1 - t) |z_1 - z_2| = (t + 1 - t) |z_1 - z_2| = |z_1 - z_2|

Thus, Option A is correct.

Analysis of Option B:

We have:
z−z1=t(z2−z1)z - z_1 = t(z_2 - z_1)
z−z2=−(1−t)(z2−z1)z - z_2 = -(1 - t)(z_2 - z_1)

Since t>0t > 0 and 1−t>01 - t > 0, the vector z−z1z - z_1 is directed from z1z_1 to zz, while z−z2z - z_2 is directed from z2z_2 to zz. These two vectors are opposite in direction, which means:
z−z1z−z2=t(z2−z1)−(1−t)(z2−z1)=−t1−t\frac{z - z_1}{z - z_2} = \frac{t(z_2 - z_1)}{-(1 - t)(z_2 - z_1)} = -\frac{t}{1 - t}
Since −t1−t-\frac{t}{1 - t} is a negative real number, the difference between their principal arguments is:
Arg(z−z1)−Arg(z−z2)=±π≠0\text{Arg}(z - z_1) - \text{Arg}(z - z_2) = \pm \pi \neq 0
  ⟹  Arg(z−z1)≠Arg(z−z2)\implies \text{Arg}(z - z_1) \neq \text{Arg}(z - z_2)

Thus, Option B is incorrect.

Analysis of Option C:

Consider the determinant:

D=∣z−z1zˉ−zˉ1z2−z1zˉ2−zˉ1∣D = \left|\begin{array}{cc} z - z_1 & \bar{z} - \bar{z}_1 \\ z_2 - z_1 & \bar{z}_2 - \bar{z}_1 \end{array}\right|

Using z−z1=t(z2−z1)z - z_1 = t(z_2 - z_1) and taking the complex conjugate (since t∈Rt \in \mathbb{R}):
zˉ−zˉ1=t(zˉ2−zˉ1)\bar{z} - \bar{z}_1 = t(\bar{z}_2 - \bar{z}_1)

Substitute these into the first row of the determinant:

D=∣t(z2−z1)t(zˉ2−zˉ1)z2−z1zˉ2−zˉ1∣D = \left|\begin{array}{cc} t(z_2 - z_1) & t(\bar{z}_2 - \bar{z}_1) \\ z_2 - z_1 & \bar{z}_2 - \bar{z}_1 \end{array}\right|

Factoring out tt from the first row:

D=t∣z2−z1zˉ2−zˉ1z2−z1zˉ2−zˉ1∣D = t \left|\begin{array}{cc} z_2 - z_1 & \bar{z}_2 - \bar{z}_1 \\ z_2 - z_1 & \bar{z}_2 - \bar{z}_1 \end{array}\right|

Since the two rows are identical, the determinant evaluates to zero:
D=0D = 0

Thus, Option C is correct.

Analysis of Option D:

Since z−z1=t(z2−z1)z - z_1 = t(z_2 - z_1) and tt is a positive real number (t>0t > 0):
Arg(z−z1)=Arg(t(z2−z1))\text{Arg}(z - z_1) = \text{Arg}(t(z_2 - z_1))
For any positive real number c>0c > 0 and nonzero complex number ww, Arg(cw)=Arg(w)\text{Arg}(cw) = \text{Arg}(w). Therefore:
Arg(z−z1)=Arg(z2−z1)\text{Arg}(z - z_1) = \text{Arg}(z_2 - z_1)

Thus, Option D is correct.

Conclusion:

The correct statements are A, C, and D.

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