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Coordinate Geometry

Descartes' revolution — merging algebra and geometry into a single powerful framework.

The Coordinate Plane

The Cartesian coordinate system, invented by René Descartes, assigns every point in the plane a unique pair (x, y). This seemingly simple idea is one of the most important in all of mathematics — it lets us use algebraic equations to describe geometric shapes.

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Distance & Midpoint

Distance: d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Midpoint: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

The distance formula is a direct consequence of the Pythagorean theorem. It generalizes to n dimensions in linear algebra: d = ‖v₁ − v₂‖.

Example: Distance between (1, 2) and (4, 6)

d = √[(4−1)² + (6−2)²] = √[9 + 16] = √25 = 5

Lines & Slope

Slope: m = (y₂ − y₁)/(x₂ − x₁) = Δy/Δx
Slope-intercept form: y = mx + b
Point-slope form: y − y₁ = m(x − x₁)
Standard form: Ax + By = C

Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals: m₁ · m₂ = −1.

The concept of slope is the geometric precursor to the derivative. In calculus, we ask: what is the slope of a curved line at a single point?

Conic Sections

The four curves obtained by cutting a cone with a plane — each has a standard equation on the coordinate plane:

Circle: (x − h)² + (y − k)² = r²
Ellipse: (x − h)²/a² + (y − k)²/b² = 1
Parabola: y = a(x − h)² + k or x = a(y − k)² + h
Hyperbola: (x − h)²/a² − (y − k)²/b² = 1

Parabolas arise in quadratic equations and physics (projectile motion). Ellipses describe planetary orbits (Kepler's first law). Hyperbolas appear in navigation systems and special relativity.

All conic sections are unified by the focus-directrix property: a conic is the set of points where the ratio of distance-to-focus / distance-to-directrix equals the eccentricity e. Circle: e = 0, Ellipse: 0 < e < 1, Parabola: e = 1, Hyperbola: e > 1. In linear algebra, conics are classified by the eigenvalues of their associated matrix.

Transformations

Geometric transformations can be expressed algebraically using coordinates:

Translation by (a, b): (x, y) → (x + a, y + b)
Reflection over x-axis: (x, y) → (x, −y)
Reflection over y-axis: (x, y) → (−x, y)

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Rotation by θ: (x, y) → (x cos θ − y sin θ, x sin θ + y cos θ) — uses trigonometry
Dilation by factor k: (x, y) → (kx, ky)

The rotation formula uses sine and cosine. In linear algebra, all these transformations are represented as matrix multiplication — an incredibly powerful unification.

Coordinate geometry is the birthplace of calculus. Newton and Leibniz asked: how do curves defined by equations change locally? The answer — derivatives and integrals — launched modern mathematics.

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