- Slides: 43
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: CENTER: (h, k) RADIUS: r (square root) (h , k) r
Example 1 h Center: (h , k ) ® ( 2 , -8 ) Radius: r 9 k r²
Example 2 Center ? Radius ?
Ellipses • Salami is often cut obliquely to obtain elliptical slices, which are larger.
Ellipses Basically, an ellipse is a squished circle Standard Equation: (h , k) a b Center: (h, k) a: major radius, length from center to edge of circle b: minor radius, length from center to top/bottom of circle * You must square root the denominator
History • 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 17 th century, Johannes Kepler eventually discovered that each planet travels around the sun in an elliptical orbit with the sun at one of its foci.
Science • 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).
Properties of Ellipses • 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.
Example 3 This must equal 1 a² b 2 Center: (-4 , 5) a: 5 b: 2
vertex Parabolas vertex Standard Equations: p>0 Opens UP Opens RIGHT p<0 Opens DOWN Opens LEFT
• One of nature's best approximations to parabolas is the path of a projectile.
• This discovery by Galileo in the 17 th 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 What is the vertex? (-2 , 5) How does it open? opens down Example 5 What is the vertex? (0 , 2) How does it open? opens right
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.
Hyperbolas Looks like: two parabolas, back to back. Center: (h , k) Standard Equations: Opens LEFT and RIGHT (h , k) Opens UP and DOWN (h , k)
Hyperbolas – Transverse Axis
Hyperbolas - Application A sonic boom shock wave has the shape of a cone, and it intersects the ground in part of a hyperbola. It hits every point on this curve at the same time, so that people in different places along the curve on the ground hear it at the same time. Because the airplane is moving forward, the hyperbolic curve moves forward and eventually the boom can be heard by everyone in its path.
Example 6 Center: (-4 , 5) Opens: Left and right
Name the conic section and its center or vertex.
Acknowledgements • http: //hotmath. com/hotmath_help/topics/parabolas. html • http: //upload. wikimedia. org/wikipedia/commons/8/85/H yperbola_(PSF). png • http: //www. funwearsports. com/NHL/CAPITALS/WCDo med. Hockey. Puck. gif • Mathwarehouse. com • http: //britton. disted. camosun. bc. ca/jbconics. htm • schools. paulding. k 12. ga. us/. . . /Introduction_to_Conics. ppt