Japan 高校3年 Mathematics C (curves and complex plane)

1. Curves and the complex plane

Parabola, ellipse, hyperbola · Parametric equations · Polar coordinates · Complex plane and polar form · De Moivre's theorem

• Conic Sections (Class 11): Circle, Parabola, Ellipse and Hyperbola – Cutting a double cone with a plane gives a circle, an ellipse, a parabola or a hyperbola; a cut through the vertex gives a point, a line or a pair of lines (degenerate conics). Circle: (x − h)² + (y − k)² = r². Parabola y² = 4ax: focus (a, 0), directrix x = −a, latus rectum 4a, e = 1. Ellipse x²/a² + y²/b² = 1 (a > b): c² = a² − b², foci (±c, 0), e = c/a < 1, latus rectum 2b²/a, PF₁ + PF₂ = 2a. Hyperbola x²/a² − y²/b² = 1: c² = a² + b², e = c/a > 1, latus rectum 2b²/a, |PF₁ − PF₂| = 2a.
• Parametric Equations: Curves, Slopes and Arc Length – In parametric form, x and y are each written using a third variable t, called the parameter. Each value of t gives one point, and as t changes the point traces a curve. You can remove t to get a Cartesian equation, find slopes with dy/dx = (dy/dt)/(dx/dt), and find speed and arc length when t is time.
• Polar Coordinates – Polar coordinates give the position of a point by its distance r from a fixed point (the pole) and the angle θ turned anticlockwise from a fixed ray (the initial line). Convert with x = r cosθ, y = r sinθ, and back with r² = x² + y², tanθ = y/x (check the quadrant). Equations r = f(θ) draw circles, cardioids, limaçons, roses and spirals. Calculus: slope dy/dx = (dy/dθ)/(dx/dθ), area = ½∫r²dθ.
• Complex Numbers and Quadratic Equations – Some equations like x² + 1 = 0 have no real answer, because no real number has a negative square. So we add a new number i with i² = −1. A complex number is z = a + ib: a is the real part and b is the imaginary part. We add, subtract and multiply complex numbers like algebra, and replace i² by −1. To divide, we multiply top and bottom by the conjugate. On the Argand plane, z = a + ib is the point (a, b). The conjugate a − ib is its mirror image in the real axis, and the modulus √(a² + b²) is its distance from the origin. Every quadratic ax² + bx + c = 0 with D = b² − 4ac < 0 has two complex roots that are conjugates of each other.