Understanding Ellipses: Definitions, Properties, and Equations
This guide explores the essential characteristics of ellipses, focusing on the geometric definition involving fixed points known as foci. It explains key concepts such as the major and minor axes, and the relationship between the foci and the center of the ellipse. Additionally, you will find the equations for both horizontal and vertical ellipses, along with step-by-step examples to find the foci, intercepts, and graphing techniques. Homework assignments reinforce understanding through practical application.
Understanding Ellipses: Definitions, Properties, and Equations
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Presentation Transcript
Ellipse: The set of all points such that the sum of the distances from a point on the ellipse to two fixed points is constant. P The points are called foci singular: focus PF1 and PF2 are called focal radii F1 F2
The center is the midpoint of the line segment joining the foci The chord passing through the foci is the major axis The chord passing through the center and perpendicular to the major axis is the minor axis F1 F2 C
y b x a -c c The equation of an ellipse: (horizontal) Where: center (h, k) sum of focal radii = 2a foci (-c, 0), (c, 0) b2 = a2 – c2
y (0,b) x (-c,0) (c,0) (a,0) Derive: b2 = a2 – c2 B F1 F2
y a c x b -c The equation of an ellipse: (vertical) Notice:a2 > b2 or c2 when: a2 is denom of “y” term → vertical when: a2 is denom of “x” term → horizontal Where: center (h, k) sum of focal radii = 2a foci (0, -c), (0, c) b2 = a2 – c2
Ex. 1: Give the x- and y-intercepts, find the foci, and graph the ellipse 9x2 + 4y2 = 144: (rewrite in correct form)
Ex. 2: Find the foci and graph the ellipse x2 + 4y2 – 16 = 0:
Ex. 4: Find an equation for the ellipse having x-intercepts 2 and y-intercepts :
Ex. 5: Find an equation for the ellipse having foci at (0, -3) and (0, 3) and sum of focal radii equal to 10:
Homework:Day 1: pg. 736 #9-27 odds (use graph paper)Day 2: worksheet (ellipse) #1-10
