Conic Sections
Ellipses – NOTES
Fill in the blanks with the following words.
center
foci
vertices
co-vertices
minor axis
major axis
An ellipse is the set of all points P in a plane such that the
sum of the distances between P and two fixed points, called
the
a
, is a constant. The line through the
foci intersects the ellipse at two
The
b
a
.
b
joins the vertices. Its
midpoint is the ellipse’s
. The
line perpendicular to the major axis at the center intersects
the ellipse at the two
, which are joined by the
.
Ellipses
or
Standard Form of the Equation:
Center:
Distance from the center to the vertices:
Distance from the center to the co-vertices:
Distance from the center to the foci:
a
b
c;
Graph the equation. Identify the center, vertices, co-vertices, and foci of the ellipse.
A.
B.
Center:
Center:
Vertices:
;
Vertices:
;
Co-vertices:
;
Co-vertices:
;
Foci:
;
Foci:
;
Write the standard form of the equation of the ellipses shown below.
C.
D.
y
y
x
5
-10
-5
5
x
-10
-5
-5
-5
-10
Complete the square to write the standard form of the equation. Then, identify the center,
vertices, co-vertices, and foci of the ellipse.
When the equation of a conic section is given in general form,
, completing
the square is the only way to convert the equation to its standard form. The steps of the process are as
follows:
1.
Rewrite the equation by grouping it the following way:
2.
Factor the leading coefficient out of the squared term (A and B) the corresponding linear term (D and E)
on each parenthesis.
3.
Find half of and . Square these. Add the squared constant on each parenthesis.
4.
Multiply the third term on each parenthesis by the leading coefficient you factored out earlier, and add
those numbers to the other side.
5.
Factor the quadratic polynomial as a perfect square trinomial. Divide the whole equation by the constant
you wrote on the right side. Simplify.
E.
.
F.
Center:
Center:
Vertices:
;
Vertices:
;
Co-vertices:
;
Co-vertices:
;
Foci:
;
Foci:
;
Name:
Date:
Per.
Conic Sections
Ellipses
Graph the equation. Identify the vertices, co-vertices, and foci of the ellipse.
1.
2.
Center:
Center:
Vertices:
;
Vertices:
;
Co-vertices:
;
Co-vertices:
;
Foci:
;
Foci:
3.
;
4.
Center:
Center:
Vertices:
;
Vertices:
;
Co-vertices:
;
Co-vertices:
;
Foci:
;
Foci:
;
Write the standard form of the equation of the ellipses shown below.
5.
6.
y
y
10
5
5
x
-5
5
x
-10
-5
5
-5
-5
7.
8.
y
y
5
5
x
-10
-5
x
5
-5
5
-5
-5
-10
Complete the square to write the standard form of the equation. Then, identify the center,
vertices, co-vertices, and foci of the ellipse.
9.
10.
Center:
Center:
Vertices:
;
Vertices:
;
Co-vertices:
;
Co-vertices:
;
Foci:
;
Foci:
;