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: 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 (h, k) a b

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. The Ellipse

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

Parabola We’ve talked about this before… a U-shaped graph 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. vertex

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 Have I seen this before? Sort of…only now we have a minus sign in the middle This equation opens up and down This equation opens left and right Center: (h, k) (h, k)