Pre-Calculus_Unit 3 Conic SectionEllipses.pptx

Pre-Calculus Unit 1.3 discusses about one of the conic sections formed by the intersection of a cone and a plane.

1. CONIC SECTION: ELLIPSE

2. Parts of the Cone

3. Overview of the Conic Sections Circle – if the cutting plane is perpendicular to the axis Ellipse – if the cutting plane is not parallel to any generator Parabola – if the cutting plane is parallel to one and only one generator Hyperbola – if the cutting plane is parallel to two generators

4. Ellipse An ellipse is the set of all coplanar points such that the sum of its distance from two fixed points is constant . The fixed points are called the foci of the ellipse.

5. Parts of the Ellipse An ellipse has 2 axes of symmetry. The longer axis is called the major axis , and the shorter axis is called the minor axis . The major axis contains the foci. Hence, the foci are inside the ellipse.

6. Parts of the Ellipse The intersection of the 2 axes is called the center of the ellipse. It is the midpoint of both the foci and the 2 vertices. The segment passing through a focus of the ellipse that is perpendicular to the major axis is called a latus rectum .

7. Characteristics of Ellipse Center: (0,0) Ellipse w/ Foci on the x-axis Ellipse w/ Foci on the y-axis Equation where and Vertices Co-vertices (0, b) ( ,0) Foci Major Axis Equation: Direction: On the x-axis Length: 2a Endpoints: Equation: x Direction: On the y-axis Length: 2a Endpoints: Minor Axis Equation: Direction: On the y-axis Length: 2b Endpoints: Equation: Direction: On the x-axis Length: 2b Endpoints: Center: (0,0) Ellipse w/ Foci on the x-axis Ellipse w/ Foci on the y-axis Equation Vertices Co-vertices Foci Major Axis Minor Axis

8. Characteristics of Ellipse Center: (0,0) Ellipse w/ Foci on the x-axis Ellipse w/ Foci on the y-axis X-intercepts Y-intercepts Directrices Latus Rectum Equation: Direction: Vertical Length: Endpoints: and Equation: Direction: Horizontal Length: Endpoints: and Permissible Values Center: (0,0) Ellipse w/ Foci on the x-axis Ellipse w/ Foci on the y-axis X-intercepts Y-intercepts Directrices Latus Rectum Permissible Values

9. Sample Exercise of Ellipse Example 1 Find the coordinates of the foci and vertices, endpoints of the major axis, minor axis, and the latus rectum for each ellipse whose center is at (0,0).  

10. Sample Exercise of Ellipse Example 2 Write the equation of the ellipse with center at the origin that satisfies the given conditions. Draw the ellipse, its foci and directrices. The foci have coordinates and a vertex at (5,0). The length of the latus rectum is and the vertices have coordinates (0,-5) and (0,5).  

11. Characteristics of Ellipse Center: ( h,k ) Ellipse w/ Foci on the x-axis Ellipse w/ Foci on the y-axis Equation where and Vertices Foci Major Axis Equation: Direction: On the x-axis Length: 2a Endpoints: Equation: x Direction: On the y-axis Length: 2a Endpoints: Minor Axis Equation: Direction: On the y-axis Length: 2b Endpoints: Equation: Direction: On the x-axis Length: 2b Endpoints: Center: ( h,k ) Ellipse w/ Foci on the x-axis Ellipse w/ Foci on the y-axis Equation Vertices Foci Major Axis Minor Axis

12. Characteristics of Ellipse Center: ( h,k ) Ellipse w/ Foci on the x-axis Ellipse w/ Foci on the y-axis X-intercepts Y-intercepts Directrices Latus Rectum Equation: Direction: Vertical Length: Endpoints: and Equation: Direction: Horizontal Length: Endpoints: and Permissible Values Center: ( h,k ) Ellipse w/ Foci on the x-axis Ellipse w/ Foci on the y-axis X-intercepts Y-intercepts Directrices Latus Rectum Permissible Values

13. Sample Exercise of Ellipse Example 1 Find the coordinates of the foci and vertices, endpoints of the major axis, minor axis, and the latus rectum for each ellipse whose center is at (h, k).  

14. Sample Exercise of Ellipse Example 2 Write the equation of the ellipse in standard form that satisfies the given conditions. Draw the ellipse, its foci and directrices. The vertices (-10,-4), (6,-4), (-2,-9) and (-2,1). foci (-7,6) and -1, 6), the sum of the distances of any point from the foci is 14