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Conic Sections and Standard Forms of Equations

Master conic sections and standard forms of equations with interactive lessons and practice problems! Designed for learners with a foundation in algebra and geometry.

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Conic Sections and Standard Forms of Equations

Study Guide

Key Definition

A conic section is the intersection of a plane and a double right circular cone. This can produce different types of conics like circles, ellipses, hyperbolas, and parabolas.

Important Notes

The general equation for any conic section is $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, where A, B, C, D, E and F are constants.
Changing the values of constants can change the shape of the conic section.
A plane passing through the vertex of the cone produces degenerate conics (a point, a line, or a pair of lines).

Mathematical Notation

$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

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

$(x - h)^2/a^2 + (y - k)^2/b^2 = 1$

$(y - k) = 4p(x - h)^2$

$(x - h)^2/a^2 - (y - k)^2/b^2 = 1$

Standard forms assume the conic is centered at (h, k) and axes are aligned with the coordinate axes.

Why It Works

Conics can be classified by the discriminant B² − 4AC from the general equation Ax² + Bxy + Cy² + Dx + Ey + F = 0. If B² − 4AC = 0, the conic is a parabola; if positive, a hyperbola; and if negative, an ellipse (and if A = C and B = 0 specifically a circle).

Remember

Conic sections are a big part of geometry and are used in various fields, including astronomy, physics, engineering and computer graphics.

Quick Reference

General Conic Equation:$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

Circle Standard Form:$(x - h)^2 + (y - k)^2 = r^2$

Ellipse Standard Form:$(x - h)^2/a^2 + (y - k)^2/b^2 = 1$

Parabola Standard Form:$(y - k) = 4p(x - h)^2$

Hyperbola Standard Form:$(x - h)^2/a^2 - (y - k)^2/b^2 = 1$

Understanding Conic Sections and Standard Forms of Equations

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Video explanation of this concept

Beginner

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Beginner Explanation

A conic section is the intersection of a plane and a cone. The four primary types are: circle, ellipse, parabola, and hyperbola, each described by its own standard equation form. For example, a circle has standard form $(x - h)^2 + (y - k)^2 = r^2$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which of the following is the correct general equation for a conic section?

Real-World Problem

Question Exercise

Intermediate

Astronomy Scenario

The comet's orbit is modeled by the equation $4x^2 + 9y^2 - 36 = 0$. What type of conic section is this?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Convert the general conic equation $2x^2 + 8x + 2y^2 - 12y + 4 = 0$ into its standard form by completing the square. Then identify the type of conic section.

Challenge Quiz

Single Choice Quiz

Advanced

If the equation of a conic section is $3x^2 + 4xy - 2y^2 = 0$, what type of conic section is it?

Recap

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