Intro to Conics

Conic Sections

Conic sections were discovered by Menaechmus, a Greek, at about 375 to 325 B.C.

He described them as those curves that could be formed by intersecting a cone with a plane.

Conic SectionsA conic section is formed when a right circular

double cone is sliced by a plane at different angles.

Notice the angles in which the plane intersects each cone.

Circle

The circle is formed when the plane sliced through the cone is ________________ to the base of the cone.

parallel

Hyperbola

The hyperbola is formed when the plane sliced through the cones is ________________ to the base of the cone.

perpendicular

Applications of Hyperbolas•The property of the hyperbola is the basis of several navigational systems used in order to locate objects.

•Hyperbolas are used in combat in sound ranging to locate the position of enemy guns by the sound of the firing of those guns.

•Some comets move in hyperbolic orbits.

•If a quantity varies inversely as another quantity, such as pressure and volume in Boyle’s Law for perfect gas (PV=k) then the graph is a hyperbola.

Parabola

The parabola is formed when the plane that cuts through the cone is parallel to the _________ of one of the cones.

side

Applications of Parabolas

The design of the headlight of a car is parabolic. The light bulb is located at the vertex of the parabolic mirror. The light beams will be reflected straight out from the headlight.

Satellite dishes capture the direct rays which are then reflected by the parabolic shape to the vertex of the dish.

Ellipse

The ellipse is formed when the plane intersecting the cone is not ______________ to the base and ____________ cut through the base.

parallel does NOT

•If a person is positioned at one focus of a room that is shaped like an ellipse (such as the Oval Office), that person can hear the soft conversation of the other people positioned at the other focus of the ellipse. The reflective nature of the ellipse sends sound or light waves from one focus of the ellipse to the other.

Applications of Ellipses

•The Lithotriptor has made it possible to rid a person of a kidney stone. A strong “explosion” is emitted from an electrode positioned at one focus of an elliptical reflector. The shock wave from that electrode bounces off the reflector and centers in on the second focus of the ellipse where the kidney stone is positioned and breaks down the kidney stone.

Why is it called a conic?

1. Circle2. Ellipse3. Parabola4. Hyperbola

1 2 3 4

One more time-Circles

• What defines it as a circle?

One more time-Ellipse

• What defines it as an ellipse

One more time-Parabola

• What defines it as a parabola?

One more time-Hyperbola

• What defines it as a hyperbola?

Symmetry – recall middle school(Shapes)

• Horizontal• Vertical• Line y=x

Symmetry – High School(Functions, equations, graphs)

• X-axis symmetry – it is reflected over the x axis

• Y-axis symmetry – it is reflected over the y axis

• Origin symmetry – it is reflected over both axis without changing shape

Do you see any symmetry?

Circle Ellipse

Do you see any symmetry?

Parabola Hyperbola

Do you see any symmetry?

Circle Ellipse

Do you see any symmetry?

Parabola Hyperbola