Introduction to Conic Sections

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

History Conic sections is one of the oldest math subject studied. The conics were discovered by Greek mathematician Menaechmus (c. 375-325 BC) Menaechmus’s intelligence was highly regarded… he tutored Alexander the Great.

Appollonius History Appollonius (c. 262-190 BC) wrote about conics in his series of books simply titled “Conic Sections”. Appollonious’ nickname was “the Great Geometer” He was the first to base the theory of all three conics on sections of one circular cone. He is also the one to give the name “ellipse”, “parabola”, and “hyperbola”.

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

Example 1 -h r² -k Center: Radius: r ( ( ) , ® h k 8 2 - , ) 9

Example 2 Center ? Radius ?

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.

The early Greek astronomers thought that the planets moved in circular orbits about an unmoving earth, since the circle is the simplest mathematical curve. In the 17th century, Johannes Kepler eventually discovered that each planet travels around the sun in an elliptical orbit with the sun at one of its foci.

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).

The ellipse has an important property that is used in the reflection of light and sound waves. Any light or signal that starts at one focus will be reflected to the other focus.

The principle is also used in the construction of "whispering galleries" such as in St. Paul's Cathedral in London. If a person whispers near one focus, he can be heard at the other focus, although he cannot be heard at many places in between.

Statuary Hall in the U.S. Capital building is elliptic.  It was in this room that John Quincy Adams, while a member of the House of Representatives, discovered this acoustical phenomenon. He situated his desk at a focal point of the elliptical ceiling, easily eavesdropping on the private conversations of other House members located near the other focal point.

The ability of the ellipse to rebound an object starting from one focus to the other focus can be demonstrated with an elliptical pool table. When a ball is placed at one focus and is thrust with a cue stick, it will rebound to the other focus. If the pool table is live enough, the ball will continue passing through each focus and rebound to the other.

Ellipse Basically an ellipse is a squished circle Center: (h , k) 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

Example 3 a² b This must equal 1 2 Center: (-4 , 5) a: 5 b: 2

We’ve talked about this before… vertex Parabola vertex 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.

One of nature's best known approximations to parabolas is the path taken by a body projected upward, as in the parabolic trajectory of a golf ball.

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.

This discovery by Galileo in the 17th century made it possible for cannoneers to work out the kind of path a cannonball would travel if it were hurtled through the air at a specific angle.

Parabolas exhibit unusual and useful reflective properties. If a light is placed at the focus of a parabolic mirror, the light will be reflected in rays parallel to its axis. In this way a straight beam of light is formed. It is for this reason that parabolic surfaces are used for headlamp reflectors. The bulb is placed at the focus for the high beam and in front of the focus for the low beam.

The opposite principle is used in the giant mirrors in reflecting telescopes and in antennas used to collect light and radio waves from outer space: ...the beam comes toward the parabolic surface and is brought into focus at the focal point.

Example 4 Example 5 opens down What is the vertex? How does it open? (-2 , 5) Example 5 opens right What is the vertex? How does it open? (0 , 2)

The Hyperbola If a right circular cone is intersected by a plane perpendicular to its axis, part of a hyperbola is formed. Such an intersection can occur in physical situations as simple as sharpening a pencil that has a polygonal cross section or in the patterns formed on a wall by a lamp shade.

What I look like…two parabolas, back to back. 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 (h , k) Center: (h , k)

Example 6 Center: (-4 , 5) Opens: Left and right

Name the conic section and its center or vertex. What am I? Name the conic section and its center or vertex.

circle; (0,0)

hyperbola; (0,0)

parabola; vertex (1,-2)

parabola; vertex (-2,-3)

circle; (2,0)

ellipse; (0,0)

hyperbola; (1,-2)

circle; (-2,-1)

hyperbola; (-5,7)

parabola; vertex (0,0)

hyperbola; (0,1)

ellipse; (-5,4)