Analytic Geometry Course
Master the full scope of analytic geometry, from the Cartesian plane to three-dimensional space and conic sections. This course builds rigorous algebraic and geometric reasoning through structured derivations, proofs, and applied problem-solving. Every major topic — lines, circles, parabolas, ellipses, and hyperbolas — is developed from first principles to advanced applications.
What you'll learn:
You will build a complete foundation in the Cartesian coordinate system and progress through linear equations, slope analysis, and parallel and perpendicular line relationships. The course covers all four conic sections — circles, parabolas, ellipses, and hyperbolas — including their standard and general equations, key features, and tangent lines. You will unify these curves through the general second-degree equation and discriminant classification. Three-dimensional coordinate geometry introduces vectors, planes, and quadric surfaces. Supplementary topics include polar coordinates, parametric equations, geometric transformations, and connections to differential and integral calculus.
How you study in practice Analytic Geometry Course
How you practise Analytic Geometry Course
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With Elevify for businesses, the course includes exercises and examples tailored to your own business and the way your company needs.
Course content
8 Chapters • 35 LessonsDuration between 4 and 360 hours (you decide)
Chapter 1HideHide detailsSee detailsFoundations of the Coordinate Plane
Foundations of the Coordinate Plane
Lesson 1 • Distance Between Two Points
Derives and applies the distance formula from the Pythagorean theorem. Connects geometric length to algebraic computation.
Lesson 2 • Midpoint Formula and Applications
Defines the midpoint as the average of coordinates and applies it to segment bisection. Reinforces coordinate arithmetic skills.
Lesson 3 • The Cartesian Coordinate System
Introduces axes, quadrants, and ordered pairs as the framework for analytic geometry. Builds the spatial vocabulary needed for all subsequent topics.
Lesson 4 • Partitioning a Line Segment
Extends midpoint concepts to dividing segments in any given ratio. Prepares students for vector and parametric representations.
Chapter 2HideHide detailsSee detailsLines in the Coordinate Plane
Lines in the Coordinate Plane
Lesson 1 • Slope and Rate of Change
Defines slope as rise over run and interprets it geometrically and contextually. Establishes slope as the central concept linking algebra and geometry.
Lesson 2 • Parallel and Perpendicular Lines
Establishes slope conditions for parallelism and perpendicularity. Students write equations of lines satisfying these conditions.
Lesson 3 • Intersection and Distance from a Point to a Line
Solves systems of two linear equations to find intersection points and derives the point-to-line distance formula. Bridges algebra and geometric measurement.
Lesson 4 • Equations of a Line
Presents slope-intercept, point-slope, and standard forms of linear equations. Students convert fluently amongst all three forms.
Lesson 5 • Angle Between Two Lines
Uses slope values and the tangent function to compute angles of inclination and intersection. Extends line analysis to trigonometric relationships.
Chapter 3HideHide detailsSee detailsThe Circle
The Circle
Lesson 1 • Standard Equation of a Circle
Derives the circle equation from the distance formula using centre and radius. Establishes the canonical form used throughout conic section study.
Lesson 2 • General Form and Completing the Square
Converts the general second-degree equation to standard form by completing the square. Students identify degenerate cases and verify real circles.
Lesson 3 • Lines and Circles: Tangents and Secants
Analyses intersections of lines with circles using substitution and the discriminant. Derives tangent line equations at a given point on a circle.
Lesson 4 • Circles Through Given Points
Determines circle equations satisfying two or three point conditions using systems of equations. Reinforces simultaneous equation skills in a geometric context.
Chapter 4HideHide detailsSee detailsThe Parabola
The Parabola
Lesson 1 • Parabolas Not Centred at the Origin
Extends standard forms to parabolas with vertex at any point using translation. Students complete the square to convert general equations.
Lesson 2 • Vertex, Axis, and Key Features
Identifies vertex, axis of symmetry, focus, directrix, and latus rectum from the equation. Builds a complete geometric picture of the parabola.
Lesson 3 • Definition and Standard Forms
Defines the parabola via the focus-directrix property and derives all four standard orientations. Connects the geometric definition to algebraic equations.
Lesson 4 • Tangent Lines and Reflective Property
Derives tangent lines to parabolas and proves the reflective property using focus and directrix. Connects analytic results to optical and engineering applications.
Chapter 5HideHide detailsSee detailsThe Ellipse
The Ellipse
Lesson 1 • Key Features of the Ellipse
Identifies vertices, co-vertices, foci, and latus rectum from the standard equation. Provides a systematic method for sketching ellipses accurately.
Lesson 2 • Translated Ellipses
Applies horizontal and vertical translations to shift ellipses to any centre. Students convert general second-degree equations to standard form.
Lesson 3 • Definition and Standard Equation
Defines the ellipse as the locus of points with constant sum of focal distances and derives its equation. Establishes semi-axes a and b and their relationship to c.
Lesson 4 • Tangent Lines and Applications
Derives tangent and normal lines to ellipses and explores the reflective property. Applies ellipse geometry to orbital and architectural contexts.
Chapter 6HideHide detailsSee detailsThe Hyperbola
The Hyperbola
Lesson 1 • Definition and Standard Equation
Defines the hyperbola as the locus with constant difference of focal distances and derives both standard orientations. Establishes the relationship amongst a, b, and c.
Lesson 2 • Translated Hyperbolas
Extends standard forms to hyperbolas centred at any point using translation. Students complete the square to identify centre, vertices, and asymptotes.
Lesson 3 • Asymptotes and Graphing
Derives asymptote equations from the standard form and uses them as graphing guides. Students sketch hyperbolas accurately using the central rectangle method.
Lesson 4 • Rectangular Hyperbola and Applications
Examines the special case where a equals b and the xy-form of the rectangular hyperbola. Applies hyperbolic geometry to navigation and physics problems.
Chapter 7HideHide detailsSee detailsConic Sections: Unified Analysis
Conic Sections: Unified Analysis
Lesson 1 • Translation and Rotation Combined
Applies both translation and rotation to reduce any conic to standard form. Reinforces systematic algebraic manipulation for complex equations.
Lesson 2 • Rotation of Axes
Eliminates the xy-term by rotating coordinate axes through a calculated angle. Students transform oblique conics into standard position for analysis.
Lesson 3 • Systems Involving Conics
Solves systems of two conic equations algebraically and graphically. Students determine the number and nature of intersection points using discriminant analysis.
Lesson 4 • Focus-Directrix Definition for All Conics
Unifies all conics through the focus-directrix ratio definition using eccentricity e. Students derive polar and Cartesian forms from this single definition.
Lesson 5 • General Second-Degree Equation
Presents the full general form Ax squared plus Bxy plus Cy squared plus Dx plus Ey plus F equals zero. Identifies each conic type by coefficient relationships.
Chapter 8HideHide detailsSee detailsAnalytic Geometry in Three Dimensions
Analytic Geometry in Three Dimensions
Lesson 1 • Quadric Surfaces
Classifies and graphs ellipsoids, paraboloids, hyperboloids, and cones using trace analysis. Extends conic section knowledge to three-dimensional surfaces.
Lesson 2 • Equations of Lines in Space
Derives parametric, symmetric, and vector forms of lines in three dimensions. Students find intersections, distances, and angles between spatial lines.
Lesson 3 • Equations of Planes
Derives the plane equation from a normal vector and a point, covering all standard forms. Students find intersections, angles, and distances involving planes.
Lesson 4 • Vectors in Three Dimensions
Defines vectors algebraically and geometrically in 3D, covering dot and cross products. Provides the computational tools needed for lines and planes.
Lesson 5 • Three-Dimensional Coordinate System
Introduces the three-dimensional Cartesian system, distance formula, and midpoint in space. Establishes the spatial framework for all 3D analytic work.
Your valid completion certificate
This course is for you:
Undergraduate students: need a rigorous bridge before entering calculus courses.
High school seniors: preparing for competitive STEM college admissions exams.
Self-taught maths enthusiasts: filling gaps left by informal or incomplete study.
Aspiring engineers: require spatial reasoning grounded in precise algebraic methods.
Career changers: moving into technical fields that demand coordinate geometry fluency.
Teachers: seeking deeper subject mastery to explain conic sections with confidence.
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