Conic Sections: Ellipses Pre–Calculus PreAP/Dual, Revised ©2015 viet.j.dang@humble.k12.tx.us 11/27/2018 3:49 AM 4.6 - Fundamental Theorem of Algebra

Shape of an ellipse from a cone 11/27/2018 3:49 AM 11.1: Ellipses

Real-Life Examples 11/27/2018 3:49 AM 11.1: Ellipses

4.6 - Fundamental Theorem of Algebra Definitions Ellipse: Set of points whose sum of the distances from two fixed points is consistent Foci: Segment point joining the vertices given at a point. It is always with the MAJOR axis Major Axis: The longer line segment of the two segments Minor Axis: The smaller line segment of the two segments Vertices: Endpoints of the major axis Co-Vertices: Endpoints of the minor axis Latus Rectum: A line segment through the foci of the shape in which it is perpendicular through the major axis and endpoints of the ellipse Eccentricity: Ratio to describe the shape of the conic 0 < e < 1 11/27/2018 3:49 AM 4.6 - Fundamental Theorem of Algebra

Formulas to know: Horizontal Axis Standard Form: If the major axis is on the x-axis (horizontal) If the major axis is on the x-axis (horizontal) a is associated with major axis. Foci of the major axis: 11/27/2018 3:49 AM 11.1: Ellipses

Formulas to know: Vertical Axis Standard Form: If the major axis is on the y-axis (vertical) If the major axis is on the y-axis (vertical) a is associated with major axis. Foci of the major axis: 11/27/2018 3:49 AM 11.1: Ellipses

All Standard Form Equations Formulas to know: All Standard Form Equations Center Length of Major Axis Length of Minor Axis Foci Equation Length of Latus Rectum Eccentricity 11/27/2018 3:49 AM 11.1: Ellipses

Brief Clip 11/27/2018 3:49 AM 11.1: Ellipses

Horizontal Ellipse (0, b) (–a, 0) (a, 0) (h, k) (0, –b) (–c, 0) F (c, 0) F (–a, 0) (a, 0) (h, k) (0, –b) Center: (h, k) Foci: (+c, 0) Length of Major Axis: 2a Vertices: (+a, 0) Length of Minor Axis: 2b Co-Vertices: (0, +b) Length of Latus Rectum: 2b2/a Latus Rectum: (+c, +b2/a) Eccentricity: c/a 11/27/2018 3:49 AM 11.1: Ellipses

Vertical Ellipse (0, a) (b, 0) (–b, 0) (h, k) F (c, 0) (b, 0) (–b, 0) (h, k) (–c, 0) F Center: (h, k) Foci: (0, +c) Length of Major Axis: 2a Vertices: (0, +a) Length of Minor Axis: 2b Co-Vertices: (+b, 0) Length of Latus Rectum: 2b2/a Latus Rectum: (+b2/a, +c) Eccentricity: c/a (0, –a) 11/27/2018 3:49 AM 11.1: Ellipses

When in trouble… GRAPH and PLOT 11/27/2018 3:49 AM 11.1: Ellipses

4.6 - Fundamental Theorem of Algebra Example 1 Graph 𝒙 𝟐 𝟐𝟓 + 𝒚 𝟐 𝟒 =𝟏 A: B: C: Type: Center: Vertices: Co-Vertices: Foci: Latus Rectum: Length of Major Axis: Length of Minor Axis: Length of Latus Rectum: Eccentricity: F F 11/27/2018 3:49 AM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Example 2 Graph 𝟏𝟔 𝒙 𝟐 +𝟒 𝒚 𝟐 =𝟔𝟒 A: B: C: Type: Center: Vertices: Co-Vertices: Foci: Latus Rectum: Length of Major Axis: Length of Minor Axis: Length of Latus Rectum: Eccentricity: F F 11/27/2018 3:49 AM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Your Turn Graph 𝒙 𝟐 𝟒𝟗 + 𝒚 𝟐 𝟖𝟏 =𝟏 A: B: C: Type: Center: Vertices: Co-Vertices: Foci: Latus Rectum: Length of Major Axis: Length of Minor Axis: Length of Latus Rectum: Eccentricity: F F 11/27/2018 3:49 AM 4.6 - Fundamental Theorem of Algebra

Steps of Writing Ellipses Equation Identify the values of A, B, and C. Plot/draw the figure with the given information if possible Write the equation and label the needed information 11/27/2018 3:49 AM 4.6 - Fundamental Theorem of Algebra

Example 3 Write an equation in standard form for an ellipse with center (0, 0), vertex at (6, 0) and co-vertex at (0, 4). Identify the values of a and b. The vertex (6, 0) gives the value of a. a = 6 The co-vertex (0, 4) gives the value of b. b = 4 11/27/2018 3:49 AM 11.1: Ellipses

Example 3 Write an equation in standard form for an ellipse with center (0, 0), vertex at (6, 0) and co-vertex at (0, 4). a = 6 b = 4 x2 a2 + = 1 y2 b2 62 42 11/27/2018 3:49 AM 11.1: Ellipses

Example 4 Write an equation in standard form for an ellipse with center (0, 0), vertices at (+5, 0) and co-vertices at (0, +3). 11/27/2018 3:49 AM 11.1: Ellipses

Your Turn Write an equation in standard form for an ellipse with center (0, 0), vertex at (0, 9) and co-vertex at (5, 0). 11/27/2018 3:49 AM 11.1: Ellipses

Example 5 An ellipse has its center located in the origin. Find the equation where the foci points are at (+8, 0), length of minor axis is 4 units long and graph. A = B = C = ?? 2 8 F F 11/27/2018 3:49 AM 11.1: Ellipses

Example 5 An ellipse has its center located in the origin. Find the equation where the foci points are at (+8, 0), length of minor axis is 4 units long and graph. A = B = C = ?? 2 8 11/27/2018 3:49 AM 11.1: Ellipses

Example 5 An ellipse has its center located in the origin. Find the equation where the foci points are at (+8, 0), length of minor axis is 4 units long and graph. A = B = C = 2√17 2 8 F F 11/27/2018 3:49 AM 11.1: Ellipses

Example 5 An ellipse has its center located in the origin. Find the equation where the foci points are at (+8, 0), length of minor axis is 4 units long and graph. A = B = C = 2√17 2 8 11/27/2018 3:49 AM 11.1: Ellipses

Example 5 An ellipse has its center located in the origin. Find the equation where the foci points are at (+8, 0), length of minor axis is 4 units long and graph. A = B = C = 2√17 2 8 11/27/2018 3:49 AM 11.1: Ellipses

Your Turn An ellipse has its center at the origin. Find an equation of the ellipse with the ends of the major axis points at (0, +5), and the foci of (0, +4). 11/27/2018 3:49 AM 11.1: Ellipses

Example 6 Graph (𝒙+𝟔) 𝟐 𝟗 + (𝒚−𝟓) 𝟐 𝟒 =𝟏 F F Type: Center: Vertices: B: C: Type: Center: Vertices: Co-Vertices: Foci: Latus Rectum: Length of Major Axis: Length of Minor Axis: Length of Latus Rectum: Eccentricity: F F 11/27/2018 3:49 AM 11.1: Ellipses

Your Turn Graph (𝒙−𝟐) 𝟐 𝟒 + (𝒚+𝟏) 𝟐 𝟏 =𝟏 F F Type: Center: Vertices: B: C: Type: Center: Vertices: Co-Vertices: Foci: Latus Rectum: Length of Major Axis: Length of Minor Axis: Length of Latus Rectum: Eccentricity: F F 11/27/2018 3:49 AM 11.1: Ellipses

4.6 - Fundamental Theorem of Algebra Example 7 An ellipse with the center of (–3, 5) has vertices at (–3, 10) and (–3, 0) and its foci is at (–3, 8) and (–3, 2). Determine the equation. F F 11/27/2018 3:49 AM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Example 8 A conic has foci points of (0, 0) and (4, 0). The major axis length is 8. Determine the equation. 11/27/2018 3:49 AM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Example 9 A conic has the vertices at (5, 0) and (5, 12). The endpoints of the minor axis are at (1, 6) and (9, 6). Determine the equation. 11/27/2018 3:49 AM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Your Turn A conic has the co-vertices at (2, 3) and (2, 1). The length of the major axis is 4. Determine the equation. 11/27/2018 3:49 AM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Standard Form Identify whether it is an ellipse by using the equation, b2 – 4ac where the answer is less than zero using the equation, Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 Rearrange variables for x’s and y’s through factoring Take GCF and add everything to other side Use completing the square; using what’s added to the x’s and y’s is added to the radius Identify the coefficient which is raised to the first power and divide the term by 2 Take the second term, divide the term by 2, and square that number Add to both sides to the equation Put the equation into factored form Put it in standard form 11/27/2018 3:49 AM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Example 10 Change the equation x2 + 4y2 + 10x – 8y + 13 = 0 to standard form. 11/27/2018 3:49 AM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Example 10 Change the equation x2 + 4y2 + 10x – 8y + 13 = 0 to standard form. 11/27/2018 3:49 AM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Example 10 Change the equation x2 + 4y2 + 10x – 8y + 13 = 0 to standard form. 11/27/2018 3:49 AM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Example 10 Change the equation x2 + 4y2 + 10x – 8y + 13 = 0 to standard form. 11/27/2018 3:49 AM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Example 11 Change the equation 4x² + 9y² – 32x + 36y + 64 = 0 to standard form. 11/27/2018 3:49 AM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Your Turn Change the equation 4x² + 9y² – 32x + 36y + 64 = 0 to standard form. 11/27/2018 3:49 AM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Assignment Worksheet 11/27/2018 3:49 AM 4.6 - Fundamental Theorem of Algebra