South 고등학교 3학년 Geometry

Chapters: 3

1. Conic sections

Parabolas · Ellipses · Hyperbolas · Conics and lines

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.

2. Plane vectors

Vectors · Vector operations · Position vectors and components · Dot product · Lines and circles with vectors

Vector Algebra – A vector has a size (magnitude) and a direction. In 3D we write it as a = xî + yĵ + zk̂. Its length is |a| = √(x² + y² + z²). Its direction cosines are l = x/|a|, m = y/|a|, n = z/|a|, and l² + m² + n² = 1. Vectors are added head-to-tail (triangle law) or component by component. ka stretches a by k and flips it if k is negative. The point dividing AB in m : n has position vector (mb + na)/(m + n) inside and (mb − na)/(m − n) outside. Dot product a·b = |a||b|cosθ gives a number and tells the angle and the projection. Cross product a×b = |a||b|sinθ n̂ gives a vector at right angles to both; its length is the area of the parallelogram on a and b.

3. Space figures and coordinates

Lines and planes in space · Three perpendiculars theorem · Orthogonal projection · 3D coordinates · Distance in space · Division of segments · Sphere equation

Three Dimensional Geometry – A line in space is fixed by one point on it and its direction. Its direction cosines l, m, n satisfy l² + m² + n² = 1; any numbers in the same ratio are direction ratios, and through two points they are (x₂ − x₁, y₂ − y₁, z₂ − z₁). Vector equation: r = a + λb. Cartesian equation: (x − x₁)/a = (y − y₁)/b = (z − z₁)/c. The angle between two lines is the angle between their directions: cosθ = |b₁·b₂|/(|b₁||b₂|). Two lines in space are parallel, intersecting or skew. The shortest distance between skew lines is d = |(a₂ − a₁)·(b₁ × b₂)|/|b₁ × b₂|; for parallel lines d = |b × (a₂ − a₁)|/|b|; d = 0 means they meet.
Solid Geometry: Points, Lines and Planes in Space – Solid geometry studies figures in three dimensions. Three points not on one line fix a plane. Two lines in space can be parallel, intersecting or skew (not in one plane). A line can lie in a plane, cut it, or be parallel to it; it is perpendicular to a plane if it is perpendicular to two intersecting lines of that plane. Angles in space are found by projecting onto a plane and using right triangles.
Introduction to Three-dimensional Geometry (Class 11) – In space we use three mutually perpendicular axes x, y, z through the origin O. Each pair makes a coordinate plane: XY (z = 0), YZ (x = 0) and ZX (y = 0). The three planes divide space into eight octants, named by the signs of x, y, z. A point P(x, y, z) is reached by moving x along the x-axis, y parallel to the y-axis and z parallel to the z-axis; x, y, z are its distances from the YZ, ZX and XY planes. Points on the x-axis are (x, 0, 0); points on the XY-plane are (x, y, 0). The distance between P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is PQ = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²).