Ellipse

1. ANALYTIC GEOMETRY
James Bernhardt P. Perpetua

2. OBJECTIVES:
Define the key terms in the Conic section such as ellipse,
eccentricity, minor and major axes, foci, vertices,
directrices and center through the discussion and
activities provided;
Solve equations and real world representations involving
ellipse through the learning activities;
Appreciate the importance of the Conic Section: Ellipse in
real-world situations and applications through
collaboration.

3. ACTIVITY 1
Materials: (a) Rubberband (b) 2 tacking pins (c) ball pen (d) piece of paper
(e) Chipboard.
Instruction:
■ Lay the paper on a flat chipboard.
■ Place the rubberband into a tacking pin.
■ Tack the pins on the paper, keeping the pins a desired distance apart.
■ Insert the ball pen point so that the string is keep taut.
■ Allow the pen to move along the inside of the string around the pins
until you get back to the starting position keeping the string taut
throughout.

4.  Move one pin farther from the other pin,
and sketch another one. What is effect on
the curve when there is more separation
between the two tacking pins?

5. ELLIPSE

6. 𝑥2
+ 𝑦2
= 1𝑥2
25
+
𝑦2
9
= 1

7. 𝑥2
25
+
𝑦2
9
= 1
𝑎2
= 25
𝑏2 = 9
𝑎 = 5
𝑏 = 3
(0, −5) (0,5)
(0, −3)
(0,3)
𝐹𝑜𝑐𝑖
𝑐2
= 𝑎2
− 𝑏2
𝑐2
= 25 − 9
𝑐2 = 16
𝑐 = 4
(0, −4) (0,4)

8. Terms:
Major axis –The line that pass through the foci.
Minor axis –The perpendicular bisector of the
major axis between the foci.
Foci – Fixed points in the ellipse.
Vertices –The points at which an ellipse
makes its sharpest turns.
Directrices –The ratio of distance of each point
from a focus to the distance of that
point from a fixed line is always the same.
Center – A point inside the ellipse which is the midpoint
of the line segment linking the two foci.
𝑎 𝑎
𝑐 𝑐𝑏
𝑏

9. Eccentricity
-The distance between its center and either of its two
foci.
𝑒 =
𝑐
𝑎

10. 𝑥2
64
+
𝑦2
25
= 1
𝑎2
= 64
𝑏2 = 25
𝑎 = 8
𝑏 = 5
(0, −5)
(0,5)
(√39,0) (√39,0)(−8,0) (8,0)
𝐹𝑜𝑐𝑖
𝑐2
= 𝑎2
− 𝑏2
𝑐2
= 64 − 25
𝑐2 = 39
𝑐 = 39

11. Quiz #1
Based on the prior activity, place what you have sketched inside the Cartesian plane.
Instructions:
1. Make a Cartesian plane on the same paper where you have sketched using
1x1 cm as the distance from one point to another. (e.g)
2. Find the center of your ellipse. Place it on the point of
origin.
3. Find the coordinates of the foci, vertices, co-vertices and their distances
from the center. (e.g. 𝑎2
=? , 𝑏2
=? 𝑎𝑛𝑑 𝑐2
=? .)

12. (𝑥 − 3)2
+(𝑦 − 2)2
= 1(𝑥 − 3)2
9
+
(𝑦 − 2)2
25
= 1

13. (𝑥 − 3)2
9
+
(𝑦 − 2)2
25
= 1
𝑎2
= 25
𝑏2 = 9
𝑎 = 5
𝑏 = 3
(3,2) (6,2)(0,2)
(3,7)
(3, −3)
𝑐2 = 𝑎2 − 𝑏2
𝑐2 = 25 − 9
𝑐2 = 16
𝑐 = 4
(3,6)
(3, −2)

14. (𝑥 − 2)2
9
+
𝑦2
16
= 1
(2,0)
𝑎2
= 16
𝑏2 = 9
𝑎 = 4
𝑏 = 3
(5,0)(−1,0)
(2,4)
(2, −4)
𝑐2 = 𝑎2 − 𝑏2
𝑐2 = 16 − 9
𝑐2
= 7
𝑐 = ± 7 (2, − 7)
(2, 7)

15. Equation
Major
Axis
Minor
Axis
Center Foci Vertices Directices
Axis of
Symmetry
𝑥2
𝑎²
+
𝑦²
𝑏²
= 1 x-axis y-axis (0,0) (±c,0) (±a,0) 𝑥 =
𝑎
𝑒
𝑎𝑛𝑑 𝑥 =
−𝑎
𝑒
both x and y
axes
𝑥2
𝑏²
+
𝑦²
𝑎²
= 1 y-axis x-axis (0,0) (0,±c) (0,±a) 𝑦 =
𝑎
𝑒
𝑎𝑛𝑑 𝑦 =
−𝑎
𝑒
both x and y
axes
Equation Foci Vertices
Co-
Vertices
Directices
Axis of
Symmetry
(𝑥 − ℎ)2
𝑎²
+
(𝑦 − 𝑘)²
𝑏²
= 1 (h±c,k) (h±a,k) (h,k±b) 𝑥 = h +
𝑎
𝑒
𝑎𝑛𝑑 𝑥 = ℎ −
𝑎
𝑒
both x and y axes
(𝑥 − ℎ)2
𝑏²
+
(𝑦 − 𝑘)²
𝑎²
= 1 (h,k±c) (h,k±a) (h±b,k) 𝑦 = 𝑘 +
𝑎
𝑒
𝑎𝑛𝑑 𝑦 = k −
𝑎
𝑒
both x and y axes
Center at origin (0,0)
Center at (h,k)

16. Instruction. Find the major axis, minor axis, eccentricity, the coordinates of
the foci and vertices, length of the major and minor axes, and the
directrices.
1. 16𝑥2
+ 9𝑦2
= 144
2. 16𝑥2
+ 25𝑦2
= 400
3. 9𝑥2
+ 𝑦2
= 9
4. 3𝑥2
+ 5𝑦2
= 15
5. 𝑥2
+ 4𝑦2
= 4

17. Instruction. Find the equation of the following.
1. foci (-1,4), (-1,6) and vertex (-1,8)
2. foci (-2,3), (-2,9) and vertex (-2,12)
3. foci (3,-1), (8,-1) and vertex (10, -1)
4. foci (4,0), (-4,0) and vertex (5,0)
5. foci (0,8), (0,-8), eccentricity 4/5