Conic Sections

Four conic sections: Circles Parabolas Ellipses Hyperbolas A plane intersecting a double cone will generate: ParabolaCircleEllipseHyperbola

Circle ©National Science Foundation

The Standard Form of a circle with a center at (h,k) and a radius, r center (3,3) radius = 2 Circle

Parabola © Art Mayoff © Long Island Fountain Company

Parabola A parabola is the set of all points in the plane that are equidistant from a fixed line (directrix) and a fixed point (focus) not on the line. Demo

Why is the focus so important?

the parabola opens : UP if p > 0, DOWN if p < 0 Vertex: (h, k) Focus: (h, k+p) Directrix:y = k - p Axis of symmetry: x = h Parabola with vertical axis of symmetry Standard equation: (x – h) 2 = 4p(y – k) p ≠ 0

Parabola with horizontal axis of symmetry the parabola opens: RIGHT if p > 0, LEFT if p < 0 Vertex: (h, k) Focus: (h + p, k) Directrix:x = h – p Axis of symmetry: y = k Standard equation:(y – k) 2 = 4p(x – h) p ≠ 0

Practice – Parabola Find the focus, vertex and directrix: 3x + 2y 2 + 8y – 4 = 0 Find the equation in standard form of a parabola with directrix x = -1 and focus (3, 2). Find the equation in standard form of a parabola with vertex at the origin and focus (5, 0).

Ellipse © Jill Britton, September 25, 2003 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.

Ellipse The ellipse is the set of all points in the plane for which the sum of distances from two fixed points (foci) is a positive constant.

Parts of an Ellipse: Major axis - longer axis, contains foci Minor axis - shorter axis Semi-axis - ½ the length of axis Center - midpoint of major axis Vertices - endpoints of the major axis Foci - two given points on the major axis Center Focus

Why are the foci of the ellipse important? 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. © Kevin Matthews and Artifice, Inc. All Rights Reserved.Artifice, Inc.

Why are the foci of the ellipse important? 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. This principle is used in lithotripsy, a medical procedure for treating kidney stones. The patient is placed in a elliptical tank of water, with the kidney stone at one focus. High-energy shock waves generated at the other focus are concentrated on the stone, pulverizing it.

Ellipse with horizontal major axis: Center is (h, k). Length of major axis is 2a. Length of minor axis is 2b. Distance between center and either focus is c with c 2 = a 2 – b 2, a > b > 0.

Ellipse with vertical major axis: Center is (h, k). Length of major axis is 2a. Length of minor axis is 2b. Distance between center and either focus is c with c 2 = a 2 – b 2, a > b > 0.

Practice – Ellipse: Graph 4x 2 + 9y 2 = 44x 2 + 9y 2 = 4 Find the vertices and foci of an ellipse - sketch the graph: 4x 2 + 9y 2 – 8x + 36y + 4 = 0 1.put in standard form 2.find center, vertices, and foci Write the equation of the ellipse with: center at (4, -2), the foci are (4, 1) and (4, -5) and the length of the minor axis is 10.

Hyperbola The huge chimney of a nuclear power plant has the shape of a hyperboloid, as does the architecture of the James S. McDonnell Planetarium of the St. Louis Science Center. © Jill Britton, September 25, 2003

Where are the Hyperbolas? 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. © Jill Britton, September 25, 2003

Hyperbola The hyperbola is the set of all points in a plane for which the difference between the distances from two fixed points (foci) is a positive constant. Differs from an Ellipse whose sum of the distances was a constant.

Parts of the hyperbola: Transverse axis Conjugate axis Vertices Foci (will be on the transverse axis) Center Asymptotes

Hyperbola with horizontal transverse axis Center is (h, k). Distance between the vertices is 2a. Distance between the foci is 2c. c 2 = a 2 + b 2

Eccentricity of the hyperbola: Eccentricity e = c/a since c > a, e >1 As the eccentricity gets larger the graph becomes wider and wider

Hyperbola with vertical transverse axis Center is (h, k). Distance between the vertices is 2b. Distance between the foci is 2c. c 2 = a 2 + b 2 + -

Practice – Hyperbola: Graph Write in standard form: 9y 2 – 25x 2 = 225 4x 2 –25y 2 +16x +50y –109 = 0 Write the equation of the hyperbolas: Vertices (0, 2) and (0, -2) Foci (0, 3) and (0, -3) Vertices (-1, 5) and (-1, -1) Foci (-1, 7) and (-1, 3)

Conclusion: Most General Equation of a Conic Section:

Which Conic is it? Parabola: A = 0 OR C = 0 Circle: A = C Ellipse: A ≠ C, but both have the same sign Hyperbola: A and C have Different signs

Example: State the type of conic and write it in the standard form of that conic Conic: A and C same sign, but A ≠ C ELLIPSE Standard Form:

Example: State the type of conic and write it in the standard form of that conic Conic: A = 0 PARABOLA Standard Form: (y – k) 2 = 4p(x – h)

Example: State the type of conic and write it in the standard form of that conic Conic: A =C CIRCLE Standard Form: **Divide all by 2!**

HW assignment - Identify, write in standard form and graph each of the following: 1)4x 2 + 9y 2 -16x - 36y -16 = 0 2)2x 2 +3y - 8x + 2 =0 3)5x - 4y =0 4)9x y x +50y = 0 5)2x 2 + 2y 2 = 10 6)(x+1) 2 + (y- 4) 2 = (x + 3) 2

Additional images