pre-calculuslesson1-conicsectionsandcircles-220911145854-d1193efe.pdf

MATH, SENIOR HIGH SCHOOL

1. PRE-CALCULUS
Ma’am Grace

2. Specific objectives:
•illustratethedifferenttypesofconicsections:parabola,ellipse,
circle,hyperbola,anddegeneratecases
•defineacircle
•determinethestandardformofequationofacircle
•graphacircleinarectangularcoordinatesystem

3. CONIC SECTIONS
Aconicsection,orsimplyconic,isacurveformedbythe
intersectionofaplaneandadoublerightcircularcone.

4. CONIC SECTIONS
Aconicsection,orsimplyconic,isacurveformedbythe
intersectionofaplaneandadoublerightcircularcone.

5. Circle 1
TYPES OF CONIC SECTIONS
Ellipse 2
Parabola 3
Hyperbola4

6. TYPES OF CONIC SECTIONS
•Circle-thecuttingplaneisnotparalleltoanygeneratorbutisperpendiculartotheaxis

7. TYPES OF CONIC SECTIONS
•Ellipse-thecuttingplaneisnotparalleltoanygenerator

8. TYPES OF CONIC SECTIONS
•Parabola-thecuttingplaneisparalleltooneandonlyonegenerator

9. TYPES OF CONIC SECTIONS
•Hyperbola-thecuttingplaneisparalleltotheaxisofthedoublecone
-isperpendiculartothebasesofthetwocones

10. TYPES OF CONIC SECTIONS
•Circle
•Ellipse
•Parabola
•Hyperbola

11. TYPES OF CONIC SECTIONS
•Circle
•Ellipse
•Parabola
•Hyperbola

12. Point
1
Line
2
Intersectinglines
3
DEGENERATE CASES:

13. DEGENERATE CASES:
•Point-thecuttingplanepassesthroughthevertexofthecone

14. DEGENERATE CASES:
•Line-thecuttingplanepassesthroughthegenerators

15. DEGENERATE CASES:
•Intersectinglines-thecuttingplanepassesthroughtheaxis

16. DEGENERATE CASES:
•Point
•Line
•Intersectinglines

17. DEGENERATE CASES:
•Point
•Line
•Intersectinglines

18. KEY ELEMENTS:
•focus (F) -the fixed point of the conic
•directrix (d) -the fixed line d corresponding to the focus
•principal axis (a) -the line that passes through the focus and
perpendicular to the directrix
•vertec (V) -the point of intersection of the conic and its principal
•eccentricity (e) -the constant ratio

19. KEY ELEMENTS:

20. CIRCLE
Acircleisasetofallcoplanarpointssuchthatthedistance
fromafixedpointisconstant.Thefixedpointiscalledthecenterof
thecicleandtheconstantdistanceformthecenteriscalledthe
radiusofthecircle.

21. EQUATION OF A CIRCLE
Given that (h, k)is the center of the circle;
and (x, y) is a pointin the circle
(x -h)
2
+ (y -k)
2
r
(x -h)
2
+ (y-k)
2
r
r
2
= (x -h)
2
+ (y-k)
2
(x-h)
2
+ (y-k)
2
= r
2
Standard form

22. EQUATION OF A CIRCLE
(x-h)
2
+ (y-k)
2
= r
2
General formx
2
+ y
2
+ Ax + By + C = 0
Standard form

23. EQUATION OF A CIRCLE
ToderivetheequationofacirclewhosecenterCisatthepoint(0,0)
andwithradiusr,letP(x,y)beoneofthepointsonthecircle.
(x -0)
2
+ (y -0)
2
r
(x -0)
2
+ (y -0)
2
r
r
2
= (x)
2
+ (y)
2
r
2
= x
2
+ y
2

24. EXAMPLES:
Determinethestandardformofequationofthecirclegivenits
centerandradius.
a.center(0,0),radius=4
b.center(2,5),radius=6
c.center(-2,7),radius=4
d.center(-8,-5),radius=3

25. EXAMPLE a.:
center(0,0),radius=4
(x-h)
2
+ (y-k)
2
= r
2

26. EXAMPLE A:
center(2,5),radius=6
(x-h)
2
+ (y-k)
2
= r
2

27. EXAMPLE B:
center(2,5),radius=6
(x-h)
2
+ (y-k)
2
= r
2
x
2
+ y
2
+ Ax + By + C = 0

28. EXAMPLE C:
center(-2,7),radius=4
(x-h)
2
+ (y-k)
2
= r
2

29. EXAMPLE D:
center(-8,-5),radius=3
(x-h)
2
+ (y-k)
2
= r
2

30. QUIZ:
Writetheequationofthecircleingeneralformandinstandard
form.
1.center(3,-2),radius=4
2.center(6,5),radius=8
3.center(0,8),radius=11
(x-h)
2
+ (y-k)
2
= r
2
x
2
+ y
2
+ Ax + By + C = 0

31. THANK YOU
May GOD bless you always!