Introduction to Conic Sections A conic section is

Introduction to Conic Sections

A conic section is a curve formed by the intersection of _____________ a plane and a double cone.

Circles The set of all points that are the same distance from the center. Standard Equation: With CENTER: (h, k) & RADIUS: r (square root) (h , k) r

Ellipse Basically an ellipse is a squished circle Standard Equation: (h , k) a b Center: (h , k) a: major radius (horizontal), length from center to edge of circle b: minor radius (vertical), length from center to top/bottom of circle * You must square root the denominator

The Ellipse • Tilt a glass of water and the surface of the liquid acquires an elliptical outline. • Salami is often cut obliquely to obtain elliptical slices which are larger.

• On a far smaller scale, the electrons of an atom move in an approximately elliptical orbit with the nucleus at one focus.

• Any cylinder sliced on an angle will reveal an ellipse in cross-section • (as seen in the Tycho Brahe Planetarium in Copenhagen).

Example This must equal 1 a² b Center: (-4 , 5) a: 5 b: 2 2

vertex Parabola We’ve talked about this before… a U-shaped graph vertex Standard Equations: OR This equation opens up or down This equation opens left or right HOW DO YOU TELL…LOOK FOR THE SQUARED VARIABLE Vertex: (h , k) • If there is a negative in front of the squared variable, then it opens down or left. • If there is NOT a negative, then it opens up or right.

• The easiest way to visualize the path of a projectile is to observe a waterspout. • Each molecule of water follows the same path and, therefore, reveals a picture of the curve.

Hyperbolas What I look like…two parabolas, back to back. Standard Equations: OR This equation opens left and right This equation opens up and down Have I seen this before? Sort of…only now we have a minus sign in the middle Center: (h , k)