Conic section

Definition

A conic section is a curve obtained by intersecting a cone with a plane. The primary types of conic sections are circles, ellipses, parabolas, and hyperbolas.

5 Must Know Facts For Your Next Test

1. The general equation for a conic section in the rotated axes form is $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.
2. Rotation of axes can eliminate the $xy$ term in the general equation of a conic section.
3. The angle of rotation to eliminate the $xy$ term can be found using $\cot(2\theta) = \frac{A - C}{B}$.
4. Ellipses and circles are characterized by having positive definite quadratic forms, while hyperbolas have indefinite quadratic forms.
5. Parabolas occur when the discriminant ($B^2 - 4AC$) equals zero.

Review Questions

• What is the effect of rotating the coordinate axes on the general equation of a conic section?
• How do you determine the angle required to eliminate the $xy$ term in a conic section's equation?
• What distinguishes an ellipse from a hyperbola in terms of their quadratic forms?

Related terms

Ellipse: A type of conic section that looks like an elongated circle. It has two foci and is defined by the equation $Ax^2 + Cy^2 = k$, where $A \neq C$ and both are positive.

Hyperbola: A type of conic section with two separate curves called branches. It is defined by an equation like $Ax^2 - Cy^2 = k$, where one coefficient is negative.

Parabola: A type of conic section that looks like an open curve. It has one focus and directrix, described by an equation such as $y = ax^2 + bx + c$.