The Kaybiang Tunnel is the longest elliptical shaped tunnel in the Philippines, connecting two towns. An ellipse is the set of all points in a plane where the sum of the distances from two fixed points, called foci, is a constant. The Kaybiang Tunnel has an elliptical shape with its longest section piercing through a mountain.

2. The Kaybiang Tunnel is
the longest elliptical
shaped tunnel in the
Philippines. It is a hole
pierced under Mt. Pico de
Loro, connecting towns of
Ternate Cavite and
Nasugbu Batangas. Locals
from nearby towns flock to
this place to enjoy the cold
weather and beautiful
scenery.

3. The Ellipses

4. Definition
An ellipse is the set of all
points in a plane such that
the distance from two fixed
points (foci) on the plane is a
constant.

5. Example of
Ellipses

9. Parts of an Ellipse
The center is (h, k)

10. Parts of an Ellipse
The center is (h, k)
𝑏
𝑎
𝑐
a distance from center to vertex
b distance from center to co−vertex
c distance from center to focus
𝑐2
= 𝑎2
− 𝑏2

11. P
R
O
P
E
R
T
I
E
S
O
F
E
L
L
I
P
S
E

12. • In all four cases above, a > b and c = 𝑎2 − 𝑏2.
• The foci F1 and F2 are c units away from the center.
• The vertices V1 and V2 are a units away from the
center, the major axis is has length 2a, the co-
vertices W1 and W2 are b units away from the
center, and the minor axis has length 2b.
• Recall that, for any point on the ellipse, the sum of its
distances from the foci is 2a.

13. In the standard equation;
•If the x-part has the bigger
denominator, the ellipse is horizontal.
•If the y-part has the bigger
denominator, the ellipse is vertical.

14. • The focus and directrix of an ellipse were considered by Pappus.
• Kepler, in 1602, said he believed that the orbit of Mars was oval,
then he later discovered that it was an ellipse with the sun at one
focus. In fact Kepler introduced the word "focus" and published
his discovery in 1609. The eccentricity of the planetary orbits is
small (i.e. they are close to circles). The eccentricity of Mars
is 1/11 and of the Earth is 1/60.
The ellipse was first studied
by Menaechmus. Euclid wrote about the ellipse
and it was given its present name by Apollonius.

15. •In 1705 Halley showed that the comet,
which is now called after him, moved in an
elliptical orbit round the sun. The
eccentricity of Halley's comet is 0.9675 so
it is close to a parabola (eccentricity 1).

16. PROPERTIES OF AN ELLIPSE
• The ellipse intersects the major axis called
𝒗𝒆𝒓𝒕𝒊𝒄𝒆𝒔, denoted as 𝑉1 and 𝑉2.
Major axis
• The length of the segment V1 V2 is equal 2a and the
length of the semi-major axis is equal to a.
𝑉1 𝑉2
Vertices

17. PROPERTIES OF AN ELLIPSE
• The ellipse intersects the major axis called
vertices, denoted as 𝑉1 and 𝑉2.
Where is the major axis?
Major axis
Minor axis
The midpoint of the segment is called
the center of the ellipse, denoted by C.

18. Eccentricity
Definition of Eccentricity: A measure of the deviation of an elliptical
path, especially an orbit, from a perfect circle. It is equal to the ratio of
the distance between the foci “2c”of the ellipse to the length of the major
axis “2a”of the ellipse (the distance between the two points farthest apart
on the ellipse).
Eccentricity ranges from zero (for a perfect circle) to values approaching 1
(highly elongated ellipses).
𝑒 =
𝑐
𝑎

19. ELLIPSE AT
C (0,0)

20. As e approaches 1
e is for eccentricity

21. 21
Foci • The two foci for this ellipse
are the two points lying on
the horizontal axis that
appear to be a little over 6
units from the origin. The
origin is the center of the
ellipse. The distance from
the center to a focus is “c”.

22. 22
Length of the Major Axis
• The segments drawn from the
two foci to the point (0,5) on
the ellipse are each 8 units in
length. Their total length is 16
units. This total length is also
the length of the major axis.

23. 23
The Ellipse
• The ends of the major axis are
at (a,0) and (-a,0).
• The ends of the minor axis are
at (0,b) and (0,-b).
• The foci are at (c,0) and (-c,0).
P(x,y)
(a,0)
(-a,0)
(0,b)
(0,-b)
(c,0)
(-c,0)
𝑥2
𝑎2
+
𝑦2
𝑏2
= 1

24. • The sum of the distances from
point P to the foci is 2a.
• Also,
P(x,y)
(a,0)
(-a,0)
(0,b)
(0,-b)
(c,0)
(-c,0)
𝑐2
= 𝑎2
− 𝑏2
or
𝑏2
= 𝑎2 − 𝑐2

25. 25
LATUS RECTUM
• A chord through a focus
and perpendicular to the
major axis is called a
latus rectum.
• The endpoints of the
two latus recti are found
using the equivalence :
±𝑐, ±
𝑏2
𝑎
𝑐2 = 𝑎2 − 𝑏2
Providing the endpoints

26. Latus Rectum
• When the equation of the ellipse
is
So the endpoints of the latus recti are:
±6.24, ±3.125
x
y
x y
𝑥2
64
+
𝑦2
25
= 1
𝑐2
= 64 − 25
= 39
6.24 ±𝑐, ±
𝑏2
𝑎

27. 27
The Ellipse
• What are the coordinates in the
endpoint major axis?
(0,13) and (0,-13).
The endpoints of the major axis are
called the vertices
• What are the coordinates in the
endpoint minor axis?
(5,0) and (-5,0)
The endpoints of the minor axis are
called the co-vertices.
𝑥2
25
+
𝑦2
169
= 1

28. 𝑐2 = 𝑎2 − 𝑏2
𝑐2
= 169 − 25
= 144
• The foci are found using
• so the values of c are
12 and -12. The
coordinates of the foci
are (0,12) and (0,-12).
𝒄𝟐
= 𝒂𝟐
− 𝒃𝟐
𝑥2
25
+
𝑦2
169
= 1

29. 29
• The endpoints of the latus recti
are:
±𝑐, ±
𝑏2
𝑎
±12, ±
25
13
𝑥2
25
+
𝑦2
169
= 1

30. Foci – the two fixed points, 𝐹1 𝑎𝑛𝑑 𝐹2, whose distances from a single
point on the ellipse is a constant.
Major axis – the line that contains the foci and goes through the center
of the ellipse.
Vertices – the two points of
intersection of the ellipse and the
major axis, 𝑉1 𝑎𝑛𝑑 𝑉2 .
Minor axis – the line that
is perpendicular to the
major axis and goes
through the center of
the ellipse.
Foci
Major axis
Vertices
Minor axis

31. Foci
Major axis
Vertices
Minor axis
𝐸1 (C,
𝑏2
𝑎
)
𝐸2 (C,
−𝑏2
𝑎
)
𝐸3 (-C,
𝑏2
𝑎
)
𝐸4 (-C,
−𝑏2
𝑎
)
𝐸1 (C,
𝑏2
𝑎
)
𝐸2 (C,
−𝑏2
𝑎
)
𝐸3 (-C,
𝑏2
𝑎
)
𝐸4 (-C,
−𝑏2
𝑎
)