Locus of Complex Numbers

Locus of Complex Numbers is obtained by letting ( z = x+yi ) and simplifying the expressions. Operations of modulus, conjugate pairs and arguments are to be used for determining the locus of complex numbers.

Question

Let \( z = x + iy \) be a complex number such that \(|z-1|^2 – |z-3|^2 = 1 \). Find the locus of all complex numbers \(z\).

\( \begin{aligned} \displaystyle \require{color}
|x+yi-1|^2-|x+yi-3|^2 &= 1 \\
\Big[(x-1)^2 + y^2\Big] – \Big[(x-3)^2 + y^2\Big] &= 1 \\
x^2 – 2x + 1 + y^2 -x^2 + 6x – 9 – y^2 &= 1 \\
4x &= 9 \\
\therefore x &= \frac{4}{9} \\
\end{aligned} \)

Algebra Algebraic Fractions Arc Binomial Expansion Capacity Chain Rule Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorials Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Proof Pythagoras Theorem Quadratic Quadratic Factorise Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume




Your email address will not be published.