A point P represents a complex number z in the Argand diagram. If ω = z / (z - 1/3i) and |ω| = 1, then the point P lies on

The correct answer is Option 4. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q7 – AP-EAMCET 2026

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Home Maths Complex & QuadraticQuestion #52

A point P represents a complex number

z

in the Argand diagram. If

ω = \frac{z}{(z - \frac{1}{3} i)}

and

| ω | = 1

, then the point P lies on

Clue 1 of 3: Geometric Interpretation of Modulus

The expression $| z_{1} | = | z_{2} |$ represents the locus of points equidistant from the points represented by $z_{1}$ and $z_{2}$ .

🧠 Full Solution Path

Step 1: Simplify the given condition✦ Active

The given condition is $| ω | = 1$ , where $ω = \frac{z}{(z - \frac{1}{3} i)}$ . Substitute the expression for $ω$ into the modulus equation.

| \frac{z}{z - \frac{1}{3} i} | = 1

Using the property $| \frac{a}{b} | = \frac{| a |}{| b |}$ , we can rewrite the equation as:

\frac{| z |}{| z - \frac{1}{3} i |} = 1

This implies:

| z | = | z - \frac{1}{3} i |

💡 Teacher's Secret Hint

Remember that $| z |$ represents the distance of the point $z$ from the origin, and $| z_{1} - z_{2} |$ represents the distance between $z_{1}$ and $z_{2}$ .

Step 2: Substitute

z = x + i y

and square both sides○ Expand

Let $z = x + i y$ , where $x$ and $y$ are real numbers. Substitute this into the equation $| z | = | z - \frac{1}{3} i |$ .

| x + i y | = | x + i y - \frac{1}{3} i |

Group the real and imaginary parts on the right side:

| x + i y | = | x + i (y - \frac{1}{3}) |

Using the definition of modulus, $| a + b i | = \sqrt{a^{2} + b^{2}}$ :

\sqrt{x^{2} + y^{2}} = \sqrt{x^{2} + {(y - \frac{1}{3})}^{2}}

Square both sides to eliminate the square roots:

x^{2} + y^{2} = x^{2} + {(y - \frac{1}{3})}^{2}

💡 Teacher's Secret Hint

Be careful with algebraic expansion, especially when dealing with fractions. $(a - b)^{2} = a^{2} - 2 a b + b^{2}$ .

Step 3: Solve for the relationship between x and y○ Expand

Subtract $x^{2}$ from both sides:

y^{2} = {(y - \frac{1}{3})}^{2}

Expand the right side:

y^{2} = y^{2} - 2 \cdot y \cdot \frac{1}{3} + {(\frac{1}{3})}^{2}

y^{2} = y^{2} - \frac{2}{3} y + \frac{1}{9}

Subtract $y^{2}$ from both sides:

0 = - \frac{2}{3} y + \frac{1}{9}

Solve for $y$ :

\frac{2}{3} y = \frac{1}{9}

y = \frac{1}{9} \cdot \frac{3}{2}

y = \frac{3}{18}

y = \frac{1}{6}

Step 4: Identify the locus○ Expand

The equation $y = \frac{1}{6}$ represents a horizontal straight line in the Cartesian coordinate system, and thus in the Argand diagram. This means the point P (representing the complex number $z$ ) lies on a straight line.

💡 Teacher's Secret Hint

Recall that in the complex plane, $z = x + i y$ . The condition $y = constant$ represents a horizontal line, while $x = constant$ represents a vertical line.

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Maths Question 7 – AP-EAMCET 2026

Clue 1 of 3: Geometric Interpretation of Modulus

🧠 Full Solution Path