Applications of Complex Numbers

• Rotation of points in the Argand plane by multiplying by e^(iθ).
• Loci problems — recognising geometric shapes from complex number equations (|z − a| = r is a circle).
• Finding roots of quadratic equations with negative discriminant using complex numbers.
• Vieta's formulas for complex roots: sum = −b/a, product = c/a.

Level 1 — Loci and quadratic roots

Loci — key forms:

• |z − z₀| = r: circle with centre z₀ and radius r.
• |z − a| = |z − b|: perpendicular bisector of segment AB.
• arg(z − z₀) = θ: ray from z₀ at angle θ.
• Re(z) = k: vertical line x = k. Im(z) = k: horizontal line y = k.

Quadratic with negative discriminant: If x² + x + 1 = 0, discriminant = 1 − 4 = −3. Roots: x = (−1 ± √(−3))/2 = −1/2 ± i√3/2. These are ω and ω².

Vieta's formulas for az² + bz + c = 0:

• Sum of roots z₁ + z₂ = −b/a
• Product of roots z₁ · z₂ = c/a
• If roots are complex conjugates α ± iβ: sum = 2α (real), product = α² + β² (real and positive).

Level 2 — Rotation, advanced loci, problem patterns

Rotation principle: To rotate point P (represented by z) about origin by angle θ, compute z·e^(iθ). To rotate about point A (represented by a), compute: z_new = a + (z − a)·e^(iθ).

Rotation formula (JEE standard): If z₁, z₂, z₃ are three complex numbers and z₂ is obtained by rotating z₁ about z₃ by angle θ: (z₂ − z₃)/(z₁ − z₃) = e^(iθ)

Argument of (z₂ − z₁)/(z₃ − z₁) = θ means the angle at z₁ between z₂ and z₃ is θ — used in circle theorems.

Locus of |z − 2|/|z + 2| = 2: Cross-multiplying and expanding gives a circle (Apollonius circle). Let z = x + iy, substitute, and simplify.

Reflection: Reflection of z across real axis is z̄. Reflection across imaginary axis is −z̄. Reflection across y = x: swap Re and Im (computed as iz̄ after adjusting).

JEE pattern — equilateral triangle: If z₁, z₂, z₃ form an equilateral triangle with centroid at origin, then z₁² + z₂² + z₃² = z₁z₂ + z₂z₃ + z₃z₁.

NCERT spotlight

Key result: If |z − 1| = |z + 1|, then Re(z) = 0 — z lies on imaginary axis (perpendicular bisector of (1,0) and (−1,0)). Argument condition arg(z) = π/4 means Im(z)/Re(z) = 1, z lies on line y = x (first quadrant ray). Complex roots of real-coefficient polynomials always come in conjugate pairs.

Find the locus of z if |z − 3i| = |z + 3i|.

Step 1 — Let z = x + iy. Then z − 3i = x + i(y − 3), z + 3i = x + i(y + 3).
Step 2 — |z − 3i|² = x² + (y−3)² and |z + 3i|² = x² + (y+3)².
Step 3 — Setting equal: x² + (y−3)² = x² + (y+3)².
Step 4 — Expand: y² − 6y + 9 = y² + 6y + 9.
Step 5 — Simplify: −6y = 6y → 12y = 0 → y = 0.
Step 6 — Locus: the real axis (Im(z) = 0). Geometric meaning: perpendicular bisector of 3i and −3i.

If z₁ = 1 + i√3, rotate z₁ by 90° anticlockwise about the origin. Find the new point z₂.

Step 1 — Rotation by 90° anticlockwise = multiply by e^(iπ/2) = i.
Step 2 — z₂ = i · z₁ = i(1 + i√3) = i + i²√3 = i − √3.
Step 3 — z₂ = −√3 + i.
Step 4 — Verify: |z₂| = √(3 + 1) = 2 = |z₁| = √(1 + 3) = 2. ✓ (Rotation preserves modulus.)
Step 5 — arg(z₁) = arctan(√3/1) = π/3. arg(z₂) = π − arctan(1/√3) = π − π/6 = 5π/6 = π/3 + π/2. ✓

1. What is the locus of z if |z − (2 + 3i)| = 4?
2. If z = 1 + i, find z after rotation by π/2 about the origin.
3. Solve z² + 4z + 8 = 0 and state the roots.
4. For roots α, ᾱ of z² − 4z + 13 = 0, find |α|.
5. Stretch: If z₁ and z₂ are two vertices of an equilateral triangle and the third vertex z₃ is obtained by rotating z₂ − z₁ by 60°, derive the formula for z₃ in terms of z₁ and z₂.

Open the Practice tab for graded questions on Complex Numbers — Applications.

• Use the platform-native live simulation or PhET-style tool for this topic (number line, Venn, physics playground, molecule builder, sensor dashboard, etc.).
• Mirror / body / home activity: physically do the concept (count objects, measure, role-play) and photograph or describe for portfolio.
• Voice or text reflection with AI Mentor: explain the concept to a younger student or family member.

• "Explain this concept to a Class 6 student using one real example from an Indian home, school, market, or festival."
• "What is one common mistake students make here, and how would you catch yourself making it?"
• Stretch: "How does this connect to coding, robotics, money, health, environment, or a future career?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.