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Chapter 4: The Ellipse
SSMth1: PrecalculusScience and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
Chapter 4: The Ellipsev Lecture 12: Introduction to Ellipsev Lecture 13: Converting General
Form to Standard Form of Ellipse and Vice-Versa
v Lecture 14: Graphing Ellipse with Center at the Origin C (0, 0)
v Lecture 15: Graphing Ellipse with Center at C (h, k)
v Lecture 16 : The Ellipse and the Tangent Line
Lecture 12: Introduction to the Ellipse
SSMth1: PrecalculusScience and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
Definition of the Ellipsev An ellipse is a set of all points
P (x, y) in a plane, the sum of whose distances from two
specified fixed points F1 and F2 is a constant.
Explain the Definition of Ellipse Using the Figure Below:
Explain the Definition of Ellipse Using the Figure Below:
Definition of the FocivThese are the two fixed
points (F1 and F2) of the ellipse.
Symbols for Foci
vF1 , F2 , to denote the foci of
the ellipse.
Definition of an Ellipsev The ellipse can also be
defined as the locus of points whose distance from
the focus is proportional to the horizontal distance from a directrix, where the ratio is
0 is less than x but x is less than 1.
Two Types of EllipsevHorizontal EllipsevVertical Ellipse
The Horizontal EllipsevWhen the foci are
on the x-axis or parallel to the x-
axis, then the ellipse is horizontal.
Something to think about…v What should be TRUEabout the coordinates of the
foci for the ellipse to be HORIZONTAL in:a) foci are on the x-axis?
b) foci are parallel to the x-axis?
Answer:a) foci are on the x-axis:
𝑥 = 𝑥|𝑥 ∈ ℝ; 𝑥 ≠ 0; 𝑖𝑓 𝑥 = 0; 𝑡ℎ𝑒𝑛 𝐹0 ≠ 𝐹1
𝑦 = 𝑦|𝑦 ∈ ℝ; 𝑦 = 0
Answer:b) foci are parallel to the x-
axis:𝑥 = 𝑥|𝑥 ∈ ℝ; 𝑥 ≠ 0; 𝑖𝑓 𝑥 = 0; 𝐹0 ≠ 𝐹1
𝑦 = 𝑦|𝑦 ∈ ℝ; 𝑦 ≠ 0
The Vertical EllipsevWhen the foci are
on the y-axis or parallel to the y-
axis, then the ellipse is vertical.
Something to think about…v What should be TRUEabout the coordinates of the
foci for the ellipse to be VERTICAL in:
a) foci are on the y-axis?b) foci are parallel to the y-axis?
Answer:a) foci are on the y-axis:
𝑥 = 𝑥|𝑥 ∈ ℝ; 𝑥 = 0
𝑦 = 𝑦|𝑦 ∈ ℝ; 𝑦 ≠ 0; 𝑖𝑓 𝑦 = 0; 𝐹0 ≠ 𝐹1
Answer:b) foci are parallel to the y-
axis:𝑥 = 𝑥|𝑥 ∈ ℝ; 𝑥 ≠ 0
𝑦 = 𝑦|𝑦 ∈ ℝ; 𝑦 ≠ 0; 𝑖𝑓 𝑦 = 0; 𝐹0 ≠ 𝐹1
The Ellipse:Parts of the Graph of the Ellipse
SSMth1: PrecalculusScience and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
Definition of the Axis of Symmetry
vThe line that passes through both foci is the axis of symmetry and
meets at two points called vertices.
Symbols for Vertices
vV1 ,V2 , to denote the vertices of the
ellipse.
Definition of the Major Axisv The line segment joining the
vertices and the foci is called the major axis. It is also
called as traverse axis and has a length of 2a.
The Semi-Major Axis
vThe letter a is called the semi-major axis of the ellipse.
Definition of the Minor Axisv The line segment which is a perpendicular bisector of the major axis is called minor axis. It is
also called as conjugate axis with a length of 2b.
The Semi-Minor Axisv The semi-minor axis is the
value of b in the length of minor axis of an ellipse
which is 2b.
Symbols for Covertices
vB1 ,B2 , to denote the co-vertices of the
ellipse.
Definition of the Center
vThe center of an ellipse is the intersection of the
major axis and the minor axis.
Definition of the Center
vC, to denote the center of the
ellipse.
Definition of the Directrixv It is a line such that the ratio
of distance of the points on the conic section from focus to its distance from the directrix is
constant.
Symbols for Directrices
vD1, D2, to denote the directrices of the
ellipse.
Definition of the Latera Recta� The plural form of latus
rectum, is the chord that passes through the focus, and is
perpendicular to the major axis and has both endpoints on the
curve.
Symbols for the Endpoints of Latera Recta
vE1, E2, E3, E4 to denote the endpoints of the
latera recta of the ellipse.
Other Symbols:
va, is the distance from the center to the vertex;
vb, is the distance from the center to one endpoint of the
minor axis;
Other Symbols:
vc, is the distance from the center to the focus;ve, is the value of the
eccentricity
Other Symbols:
v2a, is the length of the major axis; and
v2b, is the length of the minor axis.
Classroom Task 11:v Using the definition of
ellipse and parts of it, derive the standard
equation of horizontal ellipse with vertex at the
origin.
Standard Equation of the Horizontal Ellipse with Vertex at the Origin:
v The Standard Equation of the Horizontal Ellipse with Vertex at the Origin:
12
2
2
2
=+by
ax
Definition of the Ellipsev An ellipse is a set of all points
P (x, y) in a plane, the sum of whose distances from two
specified fixed points F1 and F2 is a constant.
The Horizontal Ellipse with Vertex at the Origin:
Representations:vLet:• P, be the point in the plane/ ellipse with
P(x, y); x = x and y = y;• F1, be one of the fixed points with F1 (c, 0);
x = c and y = 0;• F2, be one of the fixed points with F2 (-c, 0);
x = -c and y = 0;• PF1, be the distance from P(x, y)
and F1(c, 0);• PF2, be the distance from P(x, y) and
F2(-c, 0); and• k, be the sum of the distances of PF1 and
PF2 which is constant.
Something to think about…vWhat is the relationship
of the sum of the two sides of a triangle to its
third side?
The Relationship:v In a triangle, the sum of
the lengths of the two sides is GREATER
THAN the third side.
Standard Equation of the Horizontal Ellipse with Vertex at the Origin:
v The Standard Equation of the Horizontal Ellipse with Vertex at the Origin:
12
2
2
2
=+by
ax
The Horizontal Ellipse with Vertex at the Origin:
Standard Equation of the Vertical Ellipse with Vertex at the Origin:
v The Standard Equation of the Vertical Ellipse with Vertex at the Origin:
12
2
2
2
=+bx
ay
Ellipse in the Real World
SSMth1: PrecalculusScience and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
TED Ed Video (Nice to Know):
vLithotripsy: Ellipse in the
Medicine Field
Extracorporeal Shock Wave Lithotripsy
Medicine: Medical Lithotripsy
Arts: Elliptical Whispering Gallery
Civil Engineering: Elliptical Bridge
Aeronautical Engineering: British Spitfire
Automobile: Elliptical Gears
Naval Architecture: Racing Sailboat
Astronomy: Planetary Motion
The Ellipse:Properties of the Ellipse
SSMth1: PrecalculusScience and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
Property Number 1 of Ellipse:
vThe length of the major axis is 2a.
Property Number 2 of Ellipse:
vThe length of the minor axis is 2b.
Property Number 3 of Ellipse:
vThe length of the latusrectum is
.2
ab
Property Number 4 of Ellipse:
� The center is the intersection of the
axes.
Property Number 5 of Ellipse:
vThe endpoints of the major axis are
called the vertices.
Property Number 6 of Ellipse:
vThe endpoints of the minor axis are
called the co-vertices.
Property Number 7 of Ellipse:
� The line segment joining the vertices is called the major
axis.
Property Number 8 of Ellipse:
vThe line segment joining the co-vertices is called the minor
axis.
Property Number 9 of Ellipse:
vThe eccentricity of the ellipse is
.10 << e
Theorem 4.2: The Eccentricity (e) of the Ellipse
v The eccentricity (e) of an ellipse is the ratio of the
undirected distance between the foci to the undirected distance between
vertices; that is:.ace =
Classroom Task 12:v We say that “circle is a special type of an ellipse.” Can you prove that circle
is a special type of an ellipse?
What do we know about their radii?
Performance Task 12:vPlease download, print
and answer the “Let’s Practice 12.” Kindly work
independently.
Lecture 12: Converting General Form of the Ellipse to Its Standard
Form and Vice-Versa
SSMth1: PrecalculusScience and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
What should you expect?v This section represents how
to convert general form of ellipse to its standard form
and vice-versa.
Table 4.1: Equations of EllipseCenter
Major Axis
General Form Standard Form
(0, 0) x-axis
(0, 0) y-axis
(h, k) x-axis
(h, k) y-axis
CAFCyAx <=++ ,022
CAFEyDxCyAx
<=++++ ,022
baby
ax
>=+ ,12
2
2
2
babky
ahx
>
=-
+- ,1)()(
2
2
2
2
babx
ay
>=+ 12
2
2
2CAFAyCx >=++ 022
babhx
aky
>=-
+- 1)()(
2
2
2
2
CAFEyDxAyCx >=++++ 022
Example 32:v Convert the following general
equations to standard form:
28889 22 =+ yx
Final Answerv The standard form is:
13632
22
=+yx
Example 33:v Convert the following general
equations to standard form:
225,14925 22 =+ yx
Final Answerv The standard form is:
12549
22
=+yx
Example 34:v Convert the following general
equations to standard form:
052162443 22 =+-++ yxyx
Final Answerv The standard form is:
13)2(
4)4( 22
=-
++ yx
Conclusion 1 about the Ellipse:
vWhen the radius of the ellipse is of positive sign, then the ellipse
exists.
Example 35:v Convert the following general
equations to standard form:0110245469 22 =+-++ yxyx
Final Answerv The standard form is:
5)2(6)3(9 22 -=-++ yx
Something to think about…
vWhat can you observe on the right side of the
equation? What can you conclude?
Conclusion 2 about the Ellipse:
vWhen the radius of the ellipse is of negative sign, then the ellipse
does not exist.
Example 36:v Convert the following general
equations to standard form:0180247249 22 =++++ yxyx
Final Answerv The standard form is:
0)3(4)4(9 22 =+++ yx
Something to think about…
vWhat can you observe on the right side of the
equation? What can you conclude?
Conclusion 3 about the Ellipse:
vWhen the radius of the ellipse is of zero value,
then the ellipse will degenerate to a point.
Example 37:v Convert the following standard
form to general form:
19)3(
25)2( 22
=-
+- xy
Final Answerv The general form is:
03636150925 22 =+--+ yxyx
Example 38:v Convert the following standard
form to general form:
136)1(
100)1( 22
=+
+- yx
Final Answerv The general form is:
0464,32007210036 22 =-+-+ yxyx
Performance Task 13:vPlease download, print
and answer the “Let’s Practice 13.” Kindly work
independently.
Lecture 13: Graphing an Ellipse with Center at the Origin
SSMth1: PrecalculusScience and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
Learning Expectation:v This section presents how
to graph an ellipse and how to determine the parts of an ellipse where the center is at
the origin.
Table 4.2 : Parts of the Graph of Ellipse with Center at the Origin
Standard Equation
Foci VerticesCo-
verticesEndpoints of Latera Recta
Directrix
F1 (c, 0)
F2 (-c, 0)
V1 (a, 0)
V2 (-a, 0)
B1 (0, b)
B2 (0, -b)
F1 (0, c)
F2 (0, -c)
V1 (0, a)
V2 (0, -a)
B1 (b, 0)
B2 (-b, 0)
÷÷ø
öççè
æ--÷÷
ø
öççè
æ-
÷÷ø
öççè
æ-÷÷
ø
öççè
æ
abcE
abcE
abcE
abcE
2
4
2
3
2
2
2
1
,,
,,
cax2
±=
÷÷ø
öççè
æ--÷÷
ø
öççè
æ-
÷÷ø
öççè
æ-÷÷
ø
öççè
æ
cabEc
abE
cabEc
abE
,,
,,
2
4
2
3
2
2
2
1
cay2
±=
12
2
2
2
=+by
ax
12
2
2
2
=+bx
ay
Example 39:v Sketch and discuss the following
equation of an ellipse:
225925 22 =+ yx
Example 40:v Find the equation of the ellipse
with center at C (0, 0), length of major axis is 10 units, and a
focus at F1 (4, 0). Identify the parts of the ellipse and sketch its
graph.
Final Answer:v The equation of the ellipse is:
.1925
22
=+yx
Example 41:v Find the equation of the ellipse
with center at C (0, 0), vertices at
and eccentricity
Identify the parts of the ellipse and sketch its graph.
),0,4(± .23
=e
Final Answer:v The equation of the ellipse is:
.1416
22
=+yx
Performance Task 14:vPlease download, print
and answer the “Let’s Practice 14.” Kindly work
independently.
Lecture 14: Graphing an Ellipse with Center at C (h, k)
SSMth1: PrecalculusScience and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
Learning Expectation:vThis section presents
how to graph an ellipse and how to determine the parts of an ellipse.
Table 4.2 : Parts of the Graph of Ellipse with Center at the Origin
Standard Equation
Foci VerticesCo-
verticesEndpoints of Latera Recta
Directrix
𝑥 − ℎ 1
𝑎1 +𝑦 − 𝑘 1
𝑏1 = 1𝐹0 ℎ + 𝑐, 𝑘𝐹1 (ℎ − 𝑐, 𝑘)
𝑉0 ℎ + 𝑎, 𝑘𝑉1(ℎ − 𝑎, 𝑘)
𝐵0 ℎ, 𝑘 + 𝑏𝐵1(ℎ, 𝑘 − 𝑏)
𝐸0 ℎ + 𝑐, 𝑘 +𝑏1
𝑎
𝐸1 ℎ − 𝑐, 𝑘 +𝑏1
𝑎
𝐸@ ℎ + 𝑐, 𝑘 −𝑏1
𝑎
𝐸A ℎ − 𝑐, 𝑘 −𝑏1
𝑎
𝑥 = ℎ ±𝑎1
𝑐
𝑦− 𝑘 1
𝑎1 +𝑥 − ℎ 1
𝑏1 = 1𝐹0 ℎ , 𝑘 + 𝑐𝐹1 (ℎ, 𝑘 − 𝑐)
𝑉0 ℎ, 𝑘 + 𝑎𝑉1(ℎ, 𝑘 − 𝑎)
𝐵0 ℎ + 𝑏, 𝑘𝐵1(ℎ − 𝑏, 𝑘)
𝐸0 ℎ +𝑏1
𝑎, 𝑘 + 𝑐
𝐸1 ℎ −𝑏1
𝑎, 𝑘 + 𝑐
𝐸@ ℎ +𝑏1
𝑎 , 𝑘 − 𝑐
𝐸A ℎ −𝑏1
𝑎, 𝑘 − 𝑐
𝑦 = 𝑘 ±𝑎1
𝑐
Example 42:v Sketch and discuss the following
equation of an ellipse:
0464,32007210036 22 =-+-+ yxyx
Example 43:v Find the equation of the ellipse with center at C (-4, 7),
a focus at F1 (-4, 11) and a vertex at V2 (-4, 12). Identify
the parts of the ellipse and sketch its graph.
Final Answer:v The equation of the ellipse is:
19)4(
25)7( 22
=+
+- xy
.0616126200925 22 =+-++ yxyxor
Example 44:v Find the equation of the ellipse with
center at C (2, -3), vertices at V1 (7, -3) and V2 (-3, -3), and eccentricity of e = 3/5. Identify the
parts of the ellipse and sketch its graph.
Final Answer:v The equation of the ellipse is:
116)3(
25)2( 22
=+
+- yx
.0111150642516 22 =-+-+ yxyxor
Performance Task 15:vPlease download, print
and answer the “Let’s Practice 15.” Kindly work
independently.
Lecture 15: Tangent to the Ellipse
SSMth1: PrecalculusScience and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
Tangential to the Ellipse
vA tangent to the ellipse is a line that
touches the ellipse at just one point.
Tangent to the Ellipse
The Equation of the LinevThe equation of the line can be
determined using the formula:
222 bmamxy +±=
Example 45:v Find the equation of the
tangent to the ellipse and the line
passes at a point PT (12, 3). Sketch its graph.
364 22 =+ yx
Final Answer:vThe equation of the tangent
line is:
.01532 =-- yx
Example 46:v Find the point on the ellipse
which is the closest, and which is the farthest point from the line
Sketch the graph.
365 22 =+ yx
.03052 =+- yx
Final Answer:
v The closest point is (-4, 2) and the farthest point is
(4, -2) to the line .01532 =-- yx
Performance Task 15:vPlease download, print
and answer the “Let’s Practice 15.” Kindly work
independently.
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