Coordinate Geometry
Straight lines, circles, parabola, ellipse, hyperbola — properties and equations.
Straight lines
Slope, intercept, distance, angle between lines.
Five forms of the equation of a straight line:
- Slope-intercept: y = mx + c. m = slope, c = y-intercept.
- Point-slope: y − y₁ = m(x − x₁).
- Two-point: (y − y₁) / (x − x₁) = (y₂ − y₁) / (x₂ − x₁).
- Intercept form: x/a + y/b = 1. a = x-intercept, b = y-intercept.
- Normal form: x cos α + y sin α = p. p = perpendicular distance from origin, α = angle of normal with positive x-axis.
- General: ax + by + c = 0.
Slope m = (y₂ − y₁) / (x₂ − x₁) = tan θ, where θ is the angle the line makes with positive x-axis.
Special slopes:
- Horizontal line: m = 0.
- Vertical line: m undefined (line: x = const).
- 45° line: m = 1.
Two lines: y = m₁x + c₁ and y = m₂x + c₂.
Parallel: m₁ = m₂. Perpendicular: m₁ · m₂ = −1.
Angle θ between two lines:
tan θ = |(m₁ − m₂) / (1 + m₁m₂)|
(For perpendicular lines, denominator = 0 → θ = 90°.)
Distance from point (x₁, y₁) to line ax + by + c = 0:
d = |ax₁ + by₁ + c| / √(a² + b²)
Distance between two parallel lines ax + by + c₁ = 0 and ax + by + c₂ = 0:
d = |c₁ − c₂| / √(a² + b²)
Foot of perpendicular from (x₁, y₁) to line ax + by + c = 0:
(x − x₁) / a = (y − y₁) / b = −(ax₁ + by₁ + c) / (a² + b²)
Worked example. Find the equation of the perpendicular bisector of the segment joining (2, 3) and (6, 7).
Midpoint = (4, 5). Slope of segment = (7−3)/(6−2) = 1. Perpendicular slope = −1.
Equation: y − 5 = −1(x − 4) → y = −x + 9 → x + y = 9.
Worked example. Distance from (1, 2) to line 3x + 4y − 11 = 0:
d = |3(1) + 4(2) − 11| / √(9 + 16) = |3 + 8 − 11| / 5 = 0. (Point lies on the line!)
Circles
Standard equation, tangent, family of circles.
Conic sections
Parabola, ellipse, hyperbola — equations and properties.
A parabola is the locus of points equidistant from a fixed point (focus) and a fixed line (directrix).
Standard form: y² = 4ax (opens right, vertex at origin, focus at (a, 0), directrix x = −a).
Four orientations:
| Equation | Opens | Vertex | Focus | Directrix |
|---|---|---|---|---|
| y² = 4ax | Right | (0, 0) | (a, 0) | x = −a |
| y² = −4ax | Left | (0, 0) | (−a, 0) | x = a |
| x² = 4ay | Up | (0, 0) | (0, a) | y = −a |
| x² = −4ay | Down | (0, 0) | (0, −a) | y = a |
Parameter a is the focal length — distance from vertex to focus.
Key points and properties (for y² = 4ax):
- Axis: the x-axis (line of symmetry).
- Latus rectum: the chord through focus, perpendicular to axis. Length = 4a. Endpoints: (a, 2a) and (a, −2a).
- Eccentricity: e = 1 (defining feature of a parabola).
- Parametric form: (at², 2at) for parameter t ∈ ℝ.
- Focal chord: chord passing through focus. If endpoints are at parameters t₁ and t₂, then t₁t₂ = −1.
- Length of focal chord = a(t₁ − t₂)² = a(t + 1/t)² (using t₂ = −1/t).
- Minimum focal chord is the latus rectum (length 4a, at t = 1, −1).
Tangent at point (at², 2at):
y · t = x + at² (equation of tangent at parameter t).
Equivalent form: tangent at (x₁, y₁) on y² = 4ax: y · y₁ = 2a(x + x₁).
Normal at (at², 2at):
y = −tx + 2at + at³.
Property: Reflective. Rays parallel to the axis reflect off the parabola and converge at the focus. This is why parabolic dish antennas focus radio waves on a receiver, and headlights project parallel beams (reverse direction).
General parabola (shifted):
(y − k)² = 4a(x − h) — vertex (h, k), opens right.
Worked example. Find vertex, focus, directrix of y² = 12x.
Compare with y² = 4ax: 4a = 12 → a = 3.
- Vertex: (0, 0).
- Focus: (3, 0).
- Directrix: x = −3.
- Latus rectum length: 12 (chord at x = 3, from y = 6 to y = −6).