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 c2 = |a2 – b2| 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 c2 = |a2 – b2| and c is the distance from the center to the foci.

Ellipse - Graphing where c2 = |a2 – b2| 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

Example #1 Ellipse

Find an equation of the ellipse with foci (0, -3) and (0, 3) whose minor axis has length 4. Sketch the ellpise.

Example #2 Ellipse

Find the standard form of the equation for the ellipse whose major axis has endpoints (-2, -1) and (6, -1) and whose minor axis has length 8. Graph the ellipse.

Example #3 Ellipse

Find the center, vertices, and foci of the ellipse. Then graph.

Example #4 Ellipse

Find the center vertices and foci of the ellipse. Then graph.

Example #5 Ellipse

Find the equation of the ellipse below

Example #6 Ellipse

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.

Board Practice Ellipse

Practice Analyze and sketch the graph of Write the conic section in standard form. Then graph. Write an equation of an ellipse that satisfies the following