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3z1=5z2−2z3 Implies Collinear Points | Complex Numbers

JEE Maths question with a full step-by-step solution.

Question

If the complex numbers

z_1, z_2, z_3

satisfy

3z_1 = 5z_2 - 2z_3

, then

z_1, z_2, z_3

lie in a/on

Acircle

Bparabola

Clinecorrect

Dhyperbola

Solution

Step 1: Rearrange the relation into a linear combination equal to zero:

3z_1 = 5z_2 - 2z_3 \implies 3z_1 - 5z_2 + 2z_3 = 0

Step 2: Check the sum of the (real) coefficients:

3 + (-5) + 2 = 0

Step 3: Use the collinearity criterion. If real numbers

p_1, p_2, p_3

, not all zero, satisfy

p_1 + p_2 + p_3 = 0

and

p_1 z_1 + p_2 z_2 + p_3 z_3 = 0

, then the points

z_1, z_2, z_3

are collinear. (Reason:

p_1 z_1 + p_2 z_2 + p_3 z_3 = 0

with

p_1 = -(p_2 + p_3)

gives

p_2(z_2 - z_1) + p_3(z_3 - z_1) = 0

, so the vectors

z_2 - z_1

and

z_3 - z_1

are real scalar multiples of each other.) Step 4: Equivalently, solving for

z_1

z_1 = \frac{5z_2 - 2z_3}{3} = \frac{5z_2 - 2z_3}{5 - 2}

This is the section formula:

z_1

divides the line joining

z_2

and

z_3

externally in the ratio

5 : 2

. A point dividing a segment lies on the same line. Therefore

z_1, z_2, z_3

are collinear — they lie on a straight line. Correct answer: (3)

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