Conics

The complex slope μ of the line through z₁ and z₂. Two lines are parallel iff their complex slopes are equal; perpendicular iff μ₁ + μ₂ = 0. Note: complex slope is not the same as the real slope m = (y₁−y₂)/(x₁−x₂). The relation is m = i(1−μ)/(1+μ) when μ ≠ −1. Connects back to: Straight Lines chapter.

Three fundamental reflections. Reflection in the line y = x maps z = a+ib to iz̄ = ib+a... wait, it maps to z̄ rotated: the image is i·z̄ rotated by... actually reflection in y=x sends (a,b)→(b,a), so z→iz̄·(−i)= conjugate rotated: the image is z' where Re(z')=Im(z), Im(z')=Re(z), giving z' = i·z̄·(−i) = ... More carefully: reflection in y=x is z' = i·z̄. Reflection in y=−x is z' = −iz̄. Connects back to: Straight Lines chapter (reflection of a point in a line).

Reflection of z in the line az̄ + āz + b = 0 (b real). The formula is: z' = (−az̄ − b)/ā in the case |a|=1. General derivation: foot of perpendicular from z to the line, then double it. The perpendicular from z to the line āz+az̄+b=0 has direction ia (rotation by 90°). Connects back to: Straight Lines (foot of perpendicular, reflection).

Every point on |z−z₀|=r is written as z₀ + re^(iθ). For the unit circle |z|=1: z = e^(iθ) = cosθ+i sinθ. This is the cleanest parametric form of a circle — one parameter θ, one formula. Tangent at the point z₁ on |z−z₀|=r: Re((z−z₀)·z̄₁⁻¹·r) = r, simplifying to: z·z̄₁ + z̄·z₁ = 2r² (for circle centred at origin). Connects back to: Circles chapter.

Equation of the chord joining two points z₁, z₂ on the circle |z|=r. When z₁→z₂, this becomes the tangent at z₁: zz̄₁ + z̄z₁ = 2r², equivalently z/z₁ + z̄/z̄₁ = 2. The chord of contact from an external point z₀ is: zz̄₀ + z̄z₀ = 2r². Connects back to: Circles chapter (chord, tangent, chord of contact).

Necessary and sufficient condition for z₁, z₂, z₃ to form an equilateral triangle. Equivalently: z₁ + z₂ω + z₃ω² = 0 or z₁ + z₂ω² + z₃ω = 0, where ω is a cube root of unity — the vertices are related by 120° rotations. If the triangle is inscribed in |z|=r, its centroid is at the centre iff z₁+z₂+z₃ = 0.

The standard parametric point on the ellipse written as a complex number. Note: z ≠ re^(iθ) here — the modulus is not constant. Equivalently: z = (a/2)(e^(iθ) + e^(−iθ)·(a/2)) = ... more cleanly: Re(z)/a = cosθ, Im(z)/b = sinθ. The sum |z−z₁|+|z−z₂| = 2a connects to the locus form in M4b. Connects back to: Ellipse chapter.

The eccentric angle θ connects the ellipse to its two auxiliary circles |z|=a and |z|=b. The point P on the ellipse has x-coordinate from the outer circle and y-coordinate from the inner circle — making the parametric form (a cosθ, b sinθ) transparent. Connects back to: Ellipse chapter (auxiliary circle, eccentric angle).

The standard parametric point (ct, c/t) on xy = c² written as a complex number. The chord joining parameters t₁ and t₂ has equation: z(t₁+t₂) + z̄·t₁t₂·(t₁+t₂)/(something)... The key JEE result: if a chord of xy=c² passes through the origin, t₁t₂ = −1, so the points are z₁ and −z̄₁. Connects back to: Hyperbola chapter (rectangular hyperbola).

The parabola y²=4ax in complex form: distance from focus (a+0i) equals distance from directrix Re(z)=−a. The RHS is exactly the distance from z to the directrix. Parametric point: z = at²+2ati = a(t+i)²+... more cleanly: z = at²+2ati. Tangent at parameter t: the line z·t = at²+... or tz = at² gives the tangent form matching the Cartesian ty = x+at². Connects back to: Parabola chapter.

The parametric point (at², 2at) written as a complex number. The chord joining t₁ and t₂: Im(z) · (t₁+t₂) = 2·Re(z) + 2at₁t₂ — matching the Cartesian chord equation y(t₁+t₂) = 2x+2at₁t₂. For a focal chord: t₁t₂ = −1, so the two endpoints are at² + 2iat and a/t² − 2ia/t. Connects back to: Parabola chapter (focal chord).

z and z' are harmonic conjugates with respect to z₁ and z₂ if they divide the segment z₁z₂ internally and externally in the same ratio. Equivalently: the cross-ratio (z, z'; z₁, z₂) = −1. The harmonic conjugate of z w.r.t. z₁, z₂ can be found by the section formula with ratio m:n internally and n:m externally.

The inversion map w = r²/z̄ sends every point z ≠ 0 to its inverse point w on the same ray from origin with |z||w| = r². Key property: circles and lines map to circles or lines under inversion. A circle through the origin maps to a line; a line not through the origin maps to a circle through the origin. This is the complex form of inverse points (cn50). Connects back to: Circles chapter (inversion, radical axis).