Conic Sections and 2nd Degree Equations

What are the four types of curves formed from cutting a right circular cone?

What is the general form of a conic?

What type of curve is formed when the cutting plane is perpendicular to the axis of a circular cone?

What type of curve is formed when the cutting plane is parallel to the surface base of the cone?

What type of curve is formed when the cutting plane is parallel only to one generator and perpendicular to the base of the cone?

What type of curve is formed when the cutting plane intersects both nappes and is parallel to two generators?

What is the condition for a circle in terms of the quadratic terms?

For the equation x² + y² - 3x + 4 = 0, what type of curve does it represent?

What is the condition for an ellipse?

For the equation 3x² - 9x = -2y² - 10y + 6, what type of curve does it represent?

What is the condition for a parabola?

For the equation 2x² - 3x - y + 7 = 0, what type of curve does it represent?

What is the condition for a hyperbola?

Conic Sections Overview

Conic sections are curves formed by cutting a right circular cone.
Four types of conic sections: circle, ellipse, parabola, and hyperbola.
General form of a conic:
( A x^2 + B xy + C y^2 + D x + E y + F = 0 )

Types of Conic Sections

Circle: Formed when a plane cuts only one nappe of the cone perpendicular to the axis.
- Condition: ( B^2 - 4AC < 0 ) with ( B = 0 ) or ( A = C )
Ellipse: Formed when a plane cuts only one nappe perpendicular to the base of the cone at an angle.
- Condition: ( B^2 - 4AC < 0 ) with ( B \neq 0 ) and ( A \neq C )
Parabola: Created when a plane is parallel to one generator of the cone and perpendicular to the base.
- Condition: ( B^2 - 4AC = 0 )
Hyperbola: Formed when a plane cuts both nappes and is parallel to two generators of the cone.
- Condition: ( B^2 - 4AC > 0 ) with ( A \neq C )

Equation Examples

For the equation ( x^2 + y^2 - 3x + 4 = 0 ):
- Quadratic terms: ( A = 1 ) (for ( x^2 )), ( C = 1 ) (for ( y^2 ))
- Result: Circle (since ( B^2 - 4AC < 0 ) and ( A = C ))
For the equation ( 3x^2 - 9x = -2y^2 - 10y + 6 ):
- Quadratic terms: ( A = 3 ) (for ( x^2 )), ( C = 2 ) (for ( y^2 ))
- Result: Ellipse (since ( B^2 - 4AC < 0 ) and ( A \neq C ))
For the equation ( 2x - 3x - y + 7 = 0 ):
- Quadratic term: Only ( A = 1 ) (for ( x^2 )) is present.
- Result: Parabola (as only one quadratic term is present).

Key Formulas

Discriminant for conic sections:
- Circle: ( B^2 - 4AC < 0 ) (either ( B = 0 ) or ( A = C ))
- Ellipse: ( B^2 - 4AC < 0 ) (where ( B \neq 0 ) and ( A \neq C ))
- Parabola: ( B^2 - 4AC = 0 )
- Hyperbola: ( B^2 - 4AC > 0 ) (with ( A \neq C ))

Distinctions:

Presence of Quadratic Terms: Both ( A x^2 ) and ( C y^2 ) required for circles and ellipses.
Only one quadratic term for parabolas.
Different coefficients indicate hyperbolas.

This quiz covers the fundamental concepts of conic sections and second-degree equations. Explore the definitions of circles, ellipses, parabolas, and hyperbolas, as well as the general form of their equations. Test your understanding of their properties and classifications.