Precalculus Lesson

Slicing Cones: From Locus Definitions to Standard & General Form

Shared by A STEM Educator

11th GradeBTC Style / Thinking Classrooms50 min

Learning Objective

Students will derive and apply the standard forms of parabolas, circles, ellipses, and hyperbolas from locus definitions and general second-degree equations, identifying geometric features including foci, directrices, asymptotes, and eccentricity.

Lesson Overview

Students discover all four conics through locus definitions and cross-sectional geometry, then build algebraic fluency transforming general second-degree equations into standard form via completing the square, culminating in identifying geometric features from equations.

Materials

Vertical whiteboards or chart paper
Dry-erase markers (multiple colors)
Pencils and paper
Basic calculator or phone calculator

Scaffolded Task Progression

1Task 1

A circle is the locus of all points equidistant from a fixed center point. Use this definition: Point

P = (x, y)

is always

r

units from center

(h, k)

. Write the distance equation, then simplify it into the standard form of a circle. Identify the center and radius of

(x - 3)^{2} + (y + 2)^{2} = 25

2Task 2

A parabola is the locus of all points equidistant from a fixed point (focus) and a fixed line (directrix). Given focus

F = (0, 2)

and directrix

y = - 2

: write the locus condition

d (P, F) = d (P, directrix)

, simplify, and write the equation in standard form

x^{2} = 4 p y

. What is

p

? Where are the focus and directrix in terms of

p

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