Conic Sections By: Danielle Hayman Mrs. Guest Per. 4.

Conic SectionsConic Sections

By: Danielle HaymanBy: Danielle Hayman

Mrs. Guest Per. 4Mrs. Guest Per. 4

What are conic sections?What are conic sections?

• Conic sections are curves that are formed by the Conic sections are curves that are formed by the intersection of a cone and a plane. The four very intersection of a cone and a plane. The four very familiar conics are knows as the circle, the ellipse, familiar conics are knows as the circle, the ellipse, the parabola and the Hyperbola. The circle is made the parabola and the Hyperbola. The circle is made when the plane and the cone intersect making a when the plane and the cone intersect making a closed curve. If the plane is perpendicular to the axis closed curve. If the plane is perpendicular to the axis of the cone the conic is called an ellipse. If the plane of the cone the conic is called an ellipse. If the plane is parallel to the line of the cone the conic is called a is parallel to the line of the cone the conic is called a parabola. Then if the intersection is an open cone parabola. Then if the intersection is an open cone this conic is called a hyperbola, this plane will this conic is called a hyperbola, this plane will intersect both halves of the cone (two separate intersect both halves of the cone (two separate curves). curves).

This is called a Circle.This is called a Circle.

• You can see in the picture that the You can see in the picture that the plane is perpendicular to the axis of plane is perpendicular to the axis of the cone. the cone. Distance formulaDistance formula

2 2 22 2 2X + Y = rX + Y = r

This is called an Ellipse.This is called an Ellipse.

• An Ellipse is the set of all points (x,y)An Ellipse is the set of all points (x,y)

2 22 2XX 22 + + YY 22 = 1 = 1a ba b

This is called a parabolaThis is called a parabola

• This is the intersection of a right This is the intersection of a right circular conical surface and circular conical surface and aa plane. plane.

22Horizontal y = 4(-2)XHorizontal y = 4(-2)XFOCUS (O,P)FOCUS (O,P) (P,0)(P,0)

This is called a hyperbola.This is called a hyperbola.

• The intersection between a conical The intersection between a conical surface and a plane which cuts surface and a plane which cuts through both halves of the cone. through both halves of the cone. Which creates two separate curves. Which creates two separate curves.

2 22 2

XX 22 + + YY 22 = 1 = 1 a ba b

Conic sectionsConic sections

• Shown together.Shown together.

Another way to see conics, and you Another way to see conics, and you can also try this at home with a can also try this at home with a Styrofoam cup.Styrofoam cup.

Translated conicsTranslated conics• Point (h,k) would be the vertex that belongs to the parabola and it is Point (h,k) would be the vertex that belongs to the parabola and it is

considered the center of other conics.considered the center of other conics.

• Translated conics:Translated conics:

• 2 2 2 2 2 2

• Circle: (x - h) + (y – k) = r Circle: (x - h) + (y – k) = r • Horizontal AxisHorizontal Axis 2 2 Vertical axis Vertical axis 2 2

• Parabola: (y – k) = 4p(x – h) (x – h) = 4p (y – k)Parabola: (y – k) = 4p(x – h) (x – h) = 4p (y – k)

• 2 2 2 22 2 2 2• Ellipse: Ellipse: (x – h)(x – h)22 + (y – k) + (y – k)2 2 (x – h)(x – h)22 + (y – k) + (y – k)22

• a b a ba b a b• 2 22 2

• HyperbolaHyperbola: (x – h): (x – h)22 - (y – k) - (y – k)22 (y – k)(y – k)22 - (x – h) - (x – h)22

• aa b a bb a b

• Notice when the parabola is horizontal the y comes before the x and the Notice when the parabola is horizontal the y comes before the x and the other way around when it is vertical, this is how you tell if the parabola is other way around when it is vertical, this is how you tell if the parabola is horizontal or vertical. horizontal or vertical.

..• Parabolas, Circles, Ellipses and hyperbolas can all be formed by Parabolas, Circles, Ellipses and hyperbolas can all be formed by intersecting a plane and a double- napped cone that is why they intersecting a plane and a double- napped cone that is why they are called conic sections. are called conic sections.

• There are equations in x and y that have graphs that are not There are equations in x and y that have graphs that are not considered conics. considered conics.

• 2 22 2

• For example x + y = 0 is just one point and instead of adding that For example x + y = 0 is just one point and instead of adding that equation if you would subtract it then it would only be two equation if you would subtract it then it would only be two intersecting lines and not a conic. intersecting lines and not a conic.

• 2 22 2

• Ax + Bxy + Cy + Dx + Ey + F = 0 these type of conics are Ax + Bxy + Cy + Dx + Ey + F = 0 these type of conics are • 22

• determined by B – 4Acdetermined by B – 4Ac• 2 2

• Ellipse and Circle : B – 4AC<0 the graph is a circle if A= CEllipse and Circle : B – 4AC<0 the graph is a circle if A= C• 22

• Parabola B - 4AC = 0 Parabola B - 4AC = 0 • 22

• Hyperbola B – 4AC>0Hyperbola B – 4AC>0

Graph the Hyperbola.Graph the Hyperbola.

Find the vertices and the foci Find the vertices and the foci of the hyperbola.of the hyperbola.

How are conics used in the real world?How are conics used in the real world?

• To build things such as a sculpture at Fermi To build things such as a sculpture at Fermi National Accelerator Laboratory. You can see all National Accelerator Laboratory. You can see all the “standard conics”the “standard conics”

Conics used in real life.Conics used in real life.

• The parabola is in the McDonalds The parabola is in the McDonalds sign.sign.

Sources Sources

• For this project I usedFor this project I used• www.kn.att.com.com/wired/fil/pages/lwww.kn.att.com.com/wired/fil/pages/l

istconicsmr.htmlistconicsmr.html• www.mathworld.comwww.mathworld.com• http://math2.org/math/algebra/http://math2.org/math/algebra/

conics.htmlconics.html• www.google.comwww.google.com• Our class Algebra two bookOur class Algebra two book