Ellipse Conic Sections

Ellipse The plane can intersect one nappe of the cone at an angle to the axis resulting in an ellipse.

Ellipse - Definition An ellipse is the set of all points in a plane such that the sum of the distances from two points (foci) is a constant. d1 + d2 = a constant value.

Finding An Equation Ellipse

Ellipse - Equation To find the equation of an ellipse, let the center be at (0, 0). The vertices on the axes are at (a, 0), (-a, 0), (0, b) and (0, -b). The foci are at (c, 0) and (-c, 0).

Ellipse - Equation According to the definition. The sum of the distances from the foci to any point on the ellipse is a constant.

Ellipse - Equation The distance from the foci to the point (a, 0) is 2a. Why?

Ellipse - Equation The distance from (c, 0) to (a, 0) is the same as from (-a, 0) to (-c, 0).

Ellipse - Equation The distance from (-c, 0) to (a, 0) added to the distance from (-a, 0) to (-c, 0) is the same as going from (-a, 0) to (a, 0) which is a distance of 2a.

Ellipse - Equation Therefore, d1 + d2 = 2a. Using the distance formula,

Ellipse - Equation Simplify: Square both sides. Subtract y2 and square binomials.

Ellipse - Equation Simplify: Solve for the term with the square root. Square both sides.

Ellipse - Equation Simplify: Get x terms, y terms, and other terms together.

Ellipse - Equation Simplify: Divide both sides by a2(c2-a2)

Ellipse - Equation Change the sign and run the negative through the denominator. At this point, let’s pause and investigate a2 – c2.

Ellipse - Equation d1 + d2 must equal 2a. However, the triangle created is an isosceles triangle and d1 = d2. Therefore, d1 and d2 for the point (0, b) must both equal “a”.

Ellipse - Equation This creates a right triangle with hypotenuse of length “a” and legs of length “b” and “c”. Using the pythagorean theorem, b2 + c2 = a2.

Ellipse - Equation We now know….. and b2 + c2 = a2 b2 = a2 – c2 Substituting for a2 - c2 where c2 = |a2 – b2|

Ellipse - Equation The equation of an ellipse centered at (0, 0) is …. where a2 = b2 + c2 and c is the distance from the center to the foci. Shifting the graph over h units and up k units, the center is at (h, k) and the equation is where a2 = b2 + c2 and c is the distance from the center to the foci.

Ellipse - Graphing where a2 = b2 + c2 and c is the distance from the center to the foci. Vertices are “a” units in the x direction an “b” units in the y direction. b a a c c The foci are “c” units in the direction of the longer (major) axis. b

Ellipse Ellipse - Graphing Graph - Example #1 Ellipse Ellipse - Graphing

Ellipse - Graphing Graph:

Graph - Example #2 Ellipse

Ellipse - Graphing Graph:

Find An Equation Ellipse

Ellipse – Find An Equation Find an equation of an ellipse with foci at (-1, -3) and (5, -3). The minor axis has a length of 4.

Ellipse – Story Problem A semielliptical arch is to have a span of 100 feet. The height of the arch, at a distance 40 feet from the center is to be 100 feet. Find the height of the arch at its center.

Ellipse – Story Problem A hall 100 feet in length is to be designed into a whispering gallery. If the foci are located 25 feet from the center, how high will the ceiling be at the center?

Assignment: Wksheet #4-7**, 20-23, 33, 38, 46, 47 **Graph and find center, major vertices, minor vertices, and foci Please use graph paper!!