Standard Equation

1. ELLIPSE
PRE-CALCULUS
CONIC SECTION

2. What is Ellipse?
➢ Bounded curve which looks like a flattened circle
Applications:
● Orbits of the planets
● Moons and comets
● Space and Astronomy
● Some airplane wings

3. Activity:

4. What is Ellipse?
➢ Set of all points in a plane such that the sum of the
distances from two points (foci) is a constant.
F1(-c,0) and F2(c,0) are Foci
➢ The set of all point P(x,y) whose
distance from F1(-c,0) and from
F2(c,0) add up to a certain constant is
called Ellipse.

5. CENTER
❑ The center has coordinate (h,k)
or (0,0)
❑ The intersection between the
major and minor axis
01 FOCI
02
❑ Two fixed points inside the ellipse which
are equidistance from the center and
lays along the major axis.
❑ For any point P in the ellipse, the sum of
its distance from the foci is equal to the
length of the major axis
ELLIPSE has the following parts

6. VERTICES
❑ Points on the ellipse, collinear with
the center and foci
❑ Endpoint of the major axis and is a
units away from the center. Thus the
segment from one vertex to the other
is 2a
03 COVERTICES
04
❑ The endpoints of the minor axis,
perpendicular to the major axis
❑ The length of the segment from the
covertices is 2b
ELLIPSE has the following parts

7. LATUS RECTUM
❑ The chord that passes through a focus and
is perpendicular to the major axis.
05
ELLIPSE has the following parts

8. STANDARD EQUATION

11. Example

12. Example

13. Example

14. Example
𝒙𝟐
𝟔𝟒
+
𝒚𝟐
𝟑𝟗
= 𝟏

15. 1.
You TRY! Find the standard equation of the following
2.
3. Foci at ( ) and whose any point on it, the sum
of its distances is 16

17. Example

18. Example

19. Example

20. Find the standard equation of the following
1. Center at (-5,1), a focus at (-2,1) and a vertex at (-1,1)
2. Foci at (0,-7) and (0,3) and vertices at (0,5) and (0,-9)
3. Foci at (-2,0) and (4,0) and whose any point on it, the sum
of its distances are 8
You TRY!

21. GENERAL FORM of the Equation
𝐴𝑥2 + 𝐵𝑦2 + 𝐶𝑥 + 𝐷𝑦 = F

22. Let’s TRY!
Write the standard equation of 𝑥2
+ 𝑦2
− 8𝑥 − 6𝑦 = −21
Write the standard equation of 2𝑥2
+ 9𝑦2
+ 16𝑥 − 90𝑦 = −239
Write the standard equation of 2𝑥2 + 9𝑦2 + 16𝑥 − 90𝑦 = −239