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Understanding and Graphing Ellipses

This document defines ellipses and discusses their key properties and equations. It begins by stating the learning objectives are to define an ellipse, determine its standard and general form equations, and sketch ellipses. It then defines an ellipse as a set of points where the sum of distances from two fixed foci is a constant. The standard form equation x2/a2 + y2/b2 = 1 is provided. Several ellipse properties and the general and standard form equations for different ellipses are also explained. The document concludes by having the reader practice graphing ellipses from equations and changing equations between standard and general forms.

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Billy Jasper Domingo
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113 views5 pages

Understanding and Graphing Ellipses

This document defines ellipses and discusses their key properties and equations. It begins by stating the learning objectives are to define an ellipse, determine its standard and general form equations, and sketch ellipses. It then defines an ellipse as a set of points where the sum of distances from two fixed foci is a constant. The standard form equation x2/a2 + y2/b2 = 1 is provided. Several ellipse properties and the general and standard form equations for different ellipses are also explained. The document concludes by having the reader practice graphing ellipses from equations and changing equations between standard and general forms.

Uploaded by

Billy Jasper Domingo
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as DOCX, PDF, TXT or read online on Scribd
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DISCOVERING ELLIPSE

Learning Competencies
At the of the lesson, the learners are expected to:
 define an ellipse
 determine the equation of an ellipse in standard form and general form.
 sketch an ellipse in a rectangular coordinate system.

Warm Up
You can still remember, a circle is a graph consisting of points equidistant from a
fixed center and the Triangle Inequality involving inequality of absolute values for
any real numbers.

Kick-Off Point

(box) An ellipse is a set of all points in a plane, thee sum whose distances from the two fixed
points is a constant. The fixed points are called foci.

(image of parts of an ellipse)

Thus, the equation of an ellipse is in the form:

x2 y 2
+ =1
a2 b 2

(box) Ellipse can also be defined as the locus of points whose distance from focus is
proportional to the horizontal distance from the directrix, where the ration is less than 1.

Properties of ellipse:

1.
2.
3.
4.
5.
6.
7.
8.

(box) Theorem

General Form and Standard Form of Ellipse.


Center Major Axis General Form Standard Form
(0,0)
(h,k)
Ellipse with Vertex at Origin

Graphing the Equation of Ellipse


1. Sketch 4x2+9y2=144

Solution:

1
(4 x 2 +9 y 2=144)
144
4 x 2 9 y 2 144
+ =
144 144 144
4 x2 9 y2
+ =1
144 144

Since, a2 = 36 and b2 = 16
Vertex: (-6,0) and (6,0)
Co Vertex2: (0,-4) and (0,4)
Focus: (-4.47,0) and (4.47,0)
E1: (4.47,-2.67)
E2: (-4.47,-2.67)

a2 −a2
Directrix: x= and x=
c c
36 2 −362
x= x=
√ 20 √ 20
¿ 8.05 ¿ 8.05

c 20
Eccentricity: e= = √ =0.75
a 6

a 2 2(16)
Latus Rectum: LR= = =5.33
a 6
Major Axis: 2a = 2(6) = 12
Minor Axis: 2b = 2(4) = 8

2.

Ellipse with Vertex at (h,k)

Graphing the Equations of the Ellipse at (h,k)


Check Point

Exercise

Application:

2.
Scoreboard

A. Change the following equations to standard form.


1. 16 x 2+ 4 y 2−32 x +16 y−32=0
2. x 2+ 4 y 2=16
3. 49 2+ 9 y 2=144
4. 9 x 2+ 4 y 2−54 x+32 y−181=0

B. Solve and complete the other parts of the ellipse. Sketch the graph.

1. 36 x 2+ 100 y 2=3600
Center:
Vertex1: a:
Vertex2: b:
Foci: c:
Directrix: e:
Latus Rectum:

2. 9 x 2+ 25 y 2−36 x+150 y +36=0

Center:
Vertex1: a:
Vertex2: b:
Foci: c:
Directrix: e:
Latus Rectum:

Time-Out Task

On your math learning diary, discuss your ideas or conclusion concerning the
topics/learning you have gained and make a brochure/output/leaflet promoting famous
ellipse-like structure in your place.

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