Loci

Locus of z at constant distance r from fixed point z₀. Directly translates from the definition of a circle — no derivation needed. |z−z₀|² = r² expands to zz̄ − z₀z̄ − z̄₀z + |z₀|² = r², the general circle in complex form.

General second-degree equation in z and z̄ representing a circle. Centre = −a, radius = √(|a|²−b). Real circle requires |a|² > b. This is the complex analogue of x²+y²+2gx+2fy+c = 0, with a = g+if and b = c.

Locus of z such that its distances from z₁ and z₂ are in constant ratio k. For k ≠ 1, this is a circle (Apollonius circle). For k = 1, it degenerates to the perpendicular bisector of z₁z₂. The centre and radius can be found by squaring and expanding.

Locus of z equidistant from z₁ and z₂ — the perpendicular bisector of the segment z₁z₂. This is the k=1 case of the Apollonius circle. Squaring and expanding: Re(z(z̄₁−z̄₂)) = (|z₁|²−|z₂|²)/2, which is the equation of the bisecting line.

Every straight line in the complex plane can be written in this form. Writing a = α+iβ recovers the Cartesian form 2αx−2βy+b = 0. The line through z₁ and z₂ can also be written as Im((z−z₁)/(z₂−z₁)) = 0, i.e., (z−z₁)/(z₂−z₁) ∈ ℝ.

Equation of the unique line passing through z₁ and z₂ in symmetric form. Alternatively: the determinant form — the 3×3 determinant with rows (z, z̄, 1), (z₁, z̄₁, 1), (z₂, z̄₂, 1) equals zero. Both forms are used in locus-type JEE problems.

Locus of z such that the direction from z₀ to z makes angle α with the positive real axis. This is a ray starting at z₀ (not included) and extending to infinity. Different values of α give different rays emanating from z₀.

Locus of z from which the segment z₁z₂ subtends a constant angle α. This is an arc of a circle passing through z₁ and z₂ — the inscribed angle theorem in complex language. The full circle is obtained by taking both α and π−α. The complementary arc gives angle π−α.

Locus of z from which z₁z₂ subtends a right angle — the circle with z₁z₂ as diameter (excluding z₁ and z₂). The +π/2 and −π/2 cases give the two semicircles. Equivalent to Re((z−z₁)(z̄−z̄₂) + (z−z₂)(z̄−z̄₁)) = 0, which is the diameter-form circle.

The ratio (z−z₁)/(z−z₂) is real iff z lies on the line through z₁ and z₂. arg = 0 means z is on the same side of z₂ as z₁ (i.e., the ray from z₂ through z₁ and beyond); arg = π means z is on the opposite side. Together they give the full line.

Locus of z whose sum of distances from two fixed points z₁ (focus 1) and z₂ (focus 2) is constant 2a. This is the definition of an ellipse — stated most cleanly in complex form. The condition 2a > |z₁−z₂| ensures the locus is non-degenerate.

Locus of z whose difference of distances from z₁ and z₂ is constant 2a. The absolute value gives both branches. When 2a = 0, the locus degenerates to the perpendicular bisector. The condition 2a < |z₁−z₂| ensures a valid hyperbola.

Replacing = with < or > converts loci into regions. Key cases: r₁ < |z−z₀| < r₂ is an annular ring; α < arg(z−z₀) < β is a sector (wedge); Im(z) > 0 is the upper half-plane; |Re(z)| < a is a vertical strip. Combining conditions gives intersection of regions.