PRE – CALCULUS
First Quarter
LESSON 1: ANALYTIC GEOMETRY: KEY CONCEPTS OF CONIC SECTIONS
Learning Outcome(s): At the end of the lesson, the learner is able to:
1. Define an ellipse;
2. Determine the standard form and general form of the equation of an ellipse;
3. Identify parts of an ellipse given specific conditions;
4. Graph an ellipse in a rectangular coordinate system; and
5. Recognize the importance of ellipse into real-life situation.
Ellipse
When the plane figure is tilted and cut only
one cone to form a bounded curve then it generates
an ELLIPSE. Based on the illustration. It implies that
the distance of segments from the foci to any points
on the ellipse is equal to the length of the major axis.
Note: We can easily determine the orientation by
comparing the denominators of the two fractions, if the higher denominator is on x2 then the orientation is horizontal
and if the higher denominator is on y2 then the orientation is vertical.
Parts Definition
Center (C) This can be at the origin (0, 0) or (h, k). This is the middle inner most part of an ellipse.
Vertices (V1 & V2) The vertices are the parts on the ellipse, collinear with the center and foci. Each vertex is a
unit/s away from the center. The length of the endpoints of the vertices is called the major
axis.
Co – Vertices (W1 & W2) Each co-vertex is b unit/s away from the center. The length of the endpoints of the co-
vertices is called the minor axis.
Foci (F1 & F2) Each focus is c unit/s away from the center.
Patterns in Finding Patterns of an Ellipse
Illustrative Example
Center at the Origin
Direction: Determine the standard form of the equation of an ellipse and identify its parts. After identifying,
plot the coordinates of its part and graph the standard form of the equation of an ellipse.
1. 289x2 + 64y2 – 18496 = 0
Solution: First, transform the general form of the equation of an ellipse into standard form.
289x2 + 64y2 – 18496 = 0 Write the general form of the equation of an ellipse.
289x2 + 64y2 = 18496 Use Addition Property of Equality (Add both sides of the equation
by 18496)
2 2
x + y =1 Use Multiplication Property of Equality (Multiply both sides
1
64 289 of the equation by )
18496
Therefore, the standard form of the equation of an ellipse is x2 + y2 = 1
64 289
Second, determine the orientation of the equation of an ellipse. Since the higher denominator is 289
which is below y2 therefore the orientation is vertical. Third, identify the values of a, b and c.
a2 = 289 a = √ 289 a = 17 b2 = 64 b = √ 64 b = 8
To find c, use the formula c2 = a2 – b2
c2 = a2 – b2
c2 = 289 – 64
c2 = 225 c = √ 225 c = 15
Since the orientation is vertical, center is at the origin and the values of a, b and c are 17, 18, and 15
consecutively therefore by substituting this to the given pattern we can identify the parts of an ellipse.
PRE – CALCULUS
FIRST QUARTER
MODULE 2
NAME:_______________________________________ STRAND:_________________________ DATE:______________
TEACHER: Gladys T. Oyao, LPT
TEST 1. Identify the values of a, b and c, and the orientation of the standard form of the equation of an ellipse.
TEST II. Determine the standard form of the equation of an ellipse and identify its parts. After identifying, plot the
coordinates of its parts and graph the equations in one rectangular coordinate system then determine what real-life
picture is in the graph.
General Form 9x2 + 81y2 – 729 = 0 121x2 + 16y2 – 1936 = 0
Standard Form
a
b
c
Orientation
Center
Vertices
Co-Vertices
Foci
