Circle Blueprints

Deriving Equations from Locus Definitions

ANALYTIC ARCHITECTURE

Project No: 01-CIRC

Locus Definition

A circle is the locus of all points in a plane that are at a fixed distance (the radius, \(r\)) from a fixed point (the center, \((h, k)\)).

Student Name

Date

1

The First Blueprint: Derivation

Let \(P(x, y)\) be any point on a circle with center \(C(h, k)\) and radius \(r\). By definition, the distance from \(C\) to \(P\) must be exactly \(r\). Use the distance formula to show your derivation below.

Step 1: Apply Distance Formula

Recall: \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)

Step 2: Eliminate the Radical

Standard Equation of a Circle

After squaring both sides, we arrive at the standard form used by architects and engineers to model circular elements:

\((x - h)^2 + (y - k)^2 = r^2\)

\((h, k)\): Coordinates of the center.
\(r\): The radius of the circle.
Note: If the center is at the origin \((0, 0)\), the equation simplifies to \(x^2 + y^2 = r^2\).

2

Drafting Practice

PROJECT 1.1: Standard Form Identification ID: CIRC-STD-01

Given the following equation, identify the center and radius of the circle:

\((x + 4)^2 + (y - 7)^2 = 36\)

Center:

Radius:

Sketch Area

PROJECT 1.2: Construction from Constraints ID: CIRC-CON-02

An architect is designing a circular fountain. The center is located at \((3, -2)\) on the site map, and the fountain must pass through the point \((7, 1)\). Determine the standard equation of the fountain's edge.

1. Calculate Radius Squared (\(r^2\))

2. Final Standard Equation

// ARCHITECT'S NOTES
- Ensure coordinates are mapped correctly.
- Double-check distance calculation.
- Keep final form simplified but expanded in thought.
- Radius represents physical width in feet.

PROJECT 1.3: Implicit Geometry ID: CIRC-IMP-03

A circle is tangent to the y-axis and has its center at \((-5, 3)\). Write its equation.

Equation:

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