Ellipses On to Sec 8 2 a Geometry

Ellipses On to Sec. 8. 2 a…

Geometry of an Ellipse An ellipse is the set of all points in a plane whose distances from two fixed points in the plane have a constant sum. The fixed points are the foci (plural of focus) of the ellipse. The line through the foci is the focal axis. The point on the focal axis midway between the foci is the center. The points where the ellipse intersect its axis are the vertices of the ellipse.

Focal Axis Focus Vertex Focus Center The sum is a constant Vertex

Deriving the Equation of an Ellipse Something to notice: Equation for the two dashed distances: Use the distance formula:

Deriving the Equation of an Ellipse

Deriving the Equation of an Ellipse This equation is the standard form of the equation of an ellipse centered at the origin with the x-axis as its focal axis. When the y-axis is the focal axis? Chord – segment with endpoints on the ellipse Major Axis – chord lying on the focal axis Minor Axis – chord through the center, perp. to the focal axis The number a is the semimajor axis, and the number b is the semiminor axis.

Ellipses with Center (0, 0) Standard Equation: (0, b) Focal Axis: x-axis Foci: Vertices: a b (–a, 0) (–c, 0) c Semimajor Axis: a Semiminor Axis: b Pythagorean Relation: (0, –b) (c, 0) (a, 0)

Ellipses with Center (0, 0) (0, a) Standard Equation: Focal Axis: y-axis (0, c) a Foci: c Vertices: Semimajor Axis: a (–b, 0) b Semiminor Axis: b Pythagorean Relation: (0, –c) (0, –a) (b, 0)

Some Practice Problems… Find the vertices and the foci of the ellipse First, write in standard form: Because the larger number is under the “x” term, the focal axis is the x-axis. So… Vertices: Foci:

Some Practice Problems… Find an equation of the ellipse with foci (0, – 3) and (0, 3) whose minor axis has length 4. Sketch the ellipse and support your sketch with a grapher. The center is (0, 0) The foci are on the y-axis with c = 3 The semiminor axis is b = 4/2 = 2 Use the Pythagorean relationship to solve for a:

Some Practice Problems… Find an equation of the ellipse with foci (0, – 3) and (0, 3) whose minor axis has length 4. Sketch the ellipse and support your sketch with a grapher. Standard Form: Can we graph by hand? To check with a calculator, solve for y:

What happens when the center of an ellipse is not on the origin? (h – c, k) (h + c, k) (h, k + a) (h, k + c) (h – a, k) (h + a, k) (h, k – c) (h, k – a)

Ellipses with Center (h, k) • Standard Equation • Focal Axis • Foci • Semimajor Axis • Semiminor Axis • Vertices • Pythagorean Relation

Ellipses with Center (h, k) • Standard Equation • Focal Axis • Semimajor Axis • Foci • Semiminor Axis • Vertices • Pythagorean Relation

Guided Practice Find the standard form of the equation for the ellipse whose major axis has endpoints (– 2, – 1) and (8, – 1), and whose minor axis has length 8. How about starting with a diagram of the given info? What is the general equation? Where is the center midpoint of the major axis!!!

Guided Practice Find the standard form of the equation for the ellipse whose major axis has endpoints (– 2, – 1) and (8, – 1), and whose minor axis has length 8. Now, find the semimajor and semiminor axes: Plug all of these values back into our general equation!!!

Whiteboard Practice Find the center, vertices, and foci of the ellipse Center: (– 2, 5) Standard form: Vertices: (– 2, 12), (– 2, – 2) Foci: (– 2, 11. 325), (– 2, – 1. 325)

Whiteboard Practice Find an equation in standard form for the ellipse that satisfies the given conditions. Major axis endpoints are (– 2, – 3) and (– 2, 7), minor axis length 4 Start with a diagram… Center: Semimajor and semiminor axes: Standard Form:

Whiteboard Practice Find an equation in standard form for the ellipse that satisfies the given conditions. The foci are (– 2, 1) and (– 2, 5); the major axis endpoints are (– 2, – 1) and (– 2, 7) Start with a diagram… Center: Semimajor axis: Semiminor axis: Standard Form: