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8.4: Ellipses • Write equations of ellipses • Graph ellipses Ellipse • An ellipse is like an oval. • Every ellipse has two axes of symmetry • Called the major axis and the minor axis • The axes intersect at the center of the ellipse • The major axis is bigger than the minor axis • We use c2 = a2 – b2 to find c • a is always greater b • The equation is always equal to 1 Ellipses Chart (pg 434) Standard Form of Equation Center Direction of Major Axis Foci Length of Major Axis Length of Minor Axis x - h a 2 2 + y - k b 2 2 =1 y - k a 2 2 + x - h b 2 2 =1 (h,k) (h,k) Horizontal Vertical (h + c, k), (h – c, k) (h, k + c), (h, k – c) 2a units 2a units 2b units 2b units Example One: Graph the ellipse x - 2 4 2 + y + 5 1 2 =1 Your Turn: Graph the ellipse x + 2 81 2 + y - 5 16 2 =1 Example Two: Graph the ellipse y - 4 64 2 + x - 2 4 2 =1 Your Turn: Graph the ellipse y - 2 36 2 + x - 4 9 2 =1 Example Three: Write the equation of the ellipse in the graph: Your Turn: Write the equation of the ellipse in the graph: Example Four: Write the equation of the ellipse in the graph: Your Turn: Write the equation of the ellipse in the graph: Standard Form Find the coordinates of the center and foci and the lengths of the major and minor axes of the ellipse with equation: x2 + 4y2 + 24y = -32 Standard Form Find the coordinates of the center and foci and the lengths of the major and minor axes of the ellipse with equation: 9x2 + 6y2 – 36x + 12y = 12