Conic Sections and 2nd Degree Equations

Study Notes

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 )

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 )

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).

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 ))

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.