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

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

This is called a Circle. You can see in the picture that the plane is perpendicular to the axis of the cone. You can see in the picture that the plane is perpendicular to the axis of the cone. Distance formula X + Y = r

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 2 X 2 + Y 2 = 1 a b

This is called a parabola This is the intersection of a right circular conical surface and a plane. This is the intersection of a right circular conical surface and a plane. 2 Horizontal y = 4(-2)X FOCUS (O,P) (P,0) (P,0)

This is called a hyperbola. The intersection between a conical surface and a plane which cuts through both halves of the cone. Which creates two separate curves. The intersection between a conical surface and a plane which cuts through both halves of the cone. Which creates two separate curves. 2 2 X 2 + Y 2 = 1 a b a b

Conic sections Shown together. Shown together.

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

Translated conics Point (h,k) would be the vertex that belongs to the parabola and it is considered the center of other conics. Point (h,k) would be the vertex that belongs to the parabola and it is considered the center of other conics. Translated conics: Translated conics: Circle: (x - h) + (y – k) = r Circle: (x - h) + (y – k) = r Horizontal Axis 2 Vertical axis 2 Horizontal Axis 2 Vertical axis 2 Parabola: (y – k) = 4p(x – h) (x – h) = 4p (y – k) Parabola: (y – k) = 4p(x – h) (x – h) = 4p (y – k) Ellipse: (x – h)2 + (y – k)2 (x – h) 2 + (y – k) 2 Ellipse: (x – h)2 + (y – k)2 (x – h) 2 + (y – k) 2 a b a b a b a b Hyperbola: (x – h) 2 - (y – k) 2 (y – k) 2 - (x – h) 2 Hyperbola: (x – h) 2 - (y – k) 2 (y – k) 2 - (x – h) 2 a b a b a b a b 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 horizontal or vertical. 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 horizontal or vertical.

. Parabolas, Circles, Ellipses and hyperbolas can all be formed by intersecting a plane and a double- napped cone that is why they are called conic sections. Parabolas, Circles, Ellipses and hyperbolas can all be formed by intersecting a plane and a double- napped cone that is why they are called conic sections. There are equations in x and y that have graphs that are not considered conics. There are equations in x and y that have graphs that are not considered conics 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 intersecting lines and not a conic. 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 intersecting lines and not a conic Ax + Bxy + Cy + Dx + Ey + F = 0 these type of conics are Ax + Bxy + Cy + Dx + Ey + F = 0 these type of conics are 2 2 determined by B – 4Ac determined by B – 4Ac 2 2 Ellipse and Circle : B – 4AC<0 the graph is a circle if A= C Ellipse and Circle : B – 4AC<0 the graph is a circle if A= C 2 2 Parabola B - 4AC = 0 Parabola B - 4AC = Hyperbola B – 4AC>0 Hyperbola B – 4AC>0

Graph the Hyperbola.

Find the vertices and the foci of the hyperbola.

How are conics used in the real world? To build things such as a sculpture at Fermi National Accelerator Laboratory. You can see all the “standard conics” To build things such as a sculpture at Fermi National Accelerator Laboratory. You can see all the “standard conics”

Conics used in real life. The parabola is in the McDonalds sign. The parabola is in the McDonalds sign.

Sources For this project I used For this project I used onicsmr.html onicsmr.html onicsmr.html onicsmr.html l l Our class Algebra two book Our class Algebra two book