PRE-CALCULUS
CONIC SECTIONS
JEFFREY D. DEL MUNDO
Mathematics Teacher
Recap:
Ellipse is
determined by a
cutting plane
(slanted) that
intersects every
generator but not
perpendicular to
the axis of the cone.
Parabola
Circle
Ellipse
Hyperbola
Conic Sections
Ellipse
Session Objectives
In this fraction of the course on Pre-Calculus, STEM
students enrolled in the subject are expected to do
the following:
1.define an ellipse.
2.determine the standard form of equation of an
ellipse.
3.graph an ellipse in a rectangular coordinate
system.
The Parabola
An Ellipse is the set
of all points, P, in a plane
the sum of whose
distances from two fixed
points, F1 and F2, is
constant. The two fixed
points are called foci. The
midpoint of the segment
connecting the foci is the
center of the ellipse.
Parts of a Parabola
▷ The line through the foci intersects the
ellipse at two points, called the vertices.
▷ The line segment that joins the vertices is
the major axis. Notice that the midpoint of
the major axis is the center of the ellipse.
▷ The line segment whose endpoints are on
the ellipse and that is perpendicular to the
major axis at the center is called the minor
axis of the ellipse.
Parts of a Parabola
Standard Form of the Equation of an Ellipse
1. Find the coordinates of the foci, vertices,
and covertices for each ellipse. Then,
draw the ellipse.
a.
b.
Examples
c.
d.
𝑥2
25
𝑥2
25
𝑦2
+ =1
9
𝑦2
+
=1
169
2
2
16𝑥 + 25𝑦 − 400 = 0
289𝑥 2 + 64𝑦 2 − 18496 = 0
Examples
2. Write the equation of the ellipse with center at the
origin that satisfies the given conditions.
a.
b.
Examples
3. Write the equation of the ellipse with center at the origin
that satisfies the given conditions.
c. Vertices: (0, -10), (0, 10); Covertices: (-6, 0), (6, 0)
d. Foci: (-5, 0), (5, 0); Vertices: (-13, 0), (13, 0)
e. Foci: (0, -4), (0, 4); Vertices: (0, -7), (0, 7)
f. Foci: (0, -2), (0, 2); y-intercepts: -3, and 3
g. Length of horizontal major axis is 8; minor axis is 4.
h. Major axis vertical with length 10; length of minor axis is 4.
Seatwork #7
1. Find the coordinates of the foci, vertices, and
covertices for each ellipse. Then, draw the ellipse.
a.
b.
c.
d.
𝑥2
100
𝑥2
144
𝑦2
+ =1
36
𝑦2
+
=1
169
2
2
25𝑥 + 16𝑦 − 400 = 0
36𝑥 2 + 100𝑦 2 − 3600 = 0
Seatwork #7
2. Write the equation of the ellipse with center at the
origin that satisfies the given conditions.
a.
b.
Seatwork #7
3. Write the equation of the ellipse with center at the origin
that satisfies the given conditions.
c. Vertices: (0, -13), (0, 13); Covertices: (-12, 0), (12, 0)
d. Foci: (-9, 0), (9, 0); Vertices: (-15, 0), (15, 0)
e. Foci: (0, -8), (0, 8); Vertices: (0, -17), (0, 17)
f. Foci: (7, 0), (-7, 0); x-intercepts: -25, and 25
g. Length of horizontal major axis is 34; minor axis is 30.
h. Major axis vertical w/ length 40; length of minor axis is 32.