Precalculus – HollyerChapter 8, Lecture BAnnouncements:Quiz 4 is on ellipses and circlesHW 11 due on 9/17Popper 02 todayConference hours: Friday, 9ish to 11 sharp1Now let’s look at Ellipses and Circles**a circle is a special case ellipse with the foci and center coincident.Definition: An ellipse is the set of all points, the sum of whose distances from two fixed points is constant. Each fixed point is called a focus (plural = foci).Basic ellipses (centered at the origin)Length of the major axis: 2a. Vertices are ellipse points at the ends of this axis.Length of the minor axis: 2b.2Note: to find c:big small-The foci are on the major axis along withthe vertices.For example: graph2 219 4x y+ =horizontal or vertical?Center:Vertices:Foci:Eccentricity:3For example: graph2 2( 2) ( 1)125 49x y+ -+ =Vertical or horizontal?Center:Vertices:Foci:Eccentricity:4Popper 02, Question 1 HW5Example:Tell me everything about:2 2(x 2) y125 16-+ =Major axis is vertical or horizontal?Center?Foci? Where are they? What are their point coordinates?Vertices:Eccentricity:Sketch the graph6Popper02 , Question 2 HW and Quiz 4 #15 7Example: Tell me everything about2 225 9 10 54 44 0x y x y+ - + - =.Put it into standard form: HWLongest axis is vertical or horizontal?CenterHWFociHWVerticesHWEccentricity8Popper, Question 39Examples:A. Tell me the following about 2 25x 5y 20x 10y 20 0+ - + - =.What kind of conic section is it? Parabola, Ellipse, Circle, HyperbolaWhere is the center of the graph?Given this new definition: The focal radii are lines from each focal point that meet at an ellipse point. The sum of the focal radii is the SAME as the length of the major axis. Let’s look at why this is true:10Popper 02, Question 4 Quiz 4 Question 2011HWPut this in standard form and give the vertices:2 225 9 100 54 44 0x y x y+ - + - =12HW Building ellipses to specifications:Write the equation of an ellipse in standard form from the following facts:Horizontal major axisVertices are at (−4, −2) and (6, 2)Eccentricity: 4/513Popper 02, Question 514Popper 02, Question 615Popper 02, Question 716Popper 02, Question 8 – Vicious and Cruel Algebra17Popper Question 918Popper, Question