INTRODUCTION TO CONIC SECTIONS (BASIC CALCULUS).pdf

INTRODUCTION TO CONIC SECTIONS (BASIC CALCULUS).pdf

1. INTRODUCTION
TO CONIC
SECTIONS
PRE-CALCULUS
STEM 11

2. LEARNING OBJECTIVES
Atthe end of this lesson, you are expected to:
•illustrate the different types of conic sections: parabola,
ellipse, circle, hyperbola, and degenerate cases
•define a circle
•determine the standard form of equation of a circle
•define a parabola
•determine the standard form of equation of a parabola

3. TYPES OF
CONIC
SECTIONS
01

4. CONE
Aconeisdefinedasa
distinctive three-
dimensionalgeometric
figurewithaflatand
curvedsurfacepointed
towardsthetop.

5. DOUBLE-NAPPED CONE
Adouble-napped
conehastwo
conesconnected
atthevertex.

6. ACTIVITY: DRAW THAT CURVE

7. CONIC SECTIONS
Conicsectionsarethecurvesobtainedfromthe
intersectionbetweenadouble-nappedconeandaplane.

8. CIRCLE
Circlesareformedwhentheintersectionoftheplaneis
perpendiculartotheaxisofrevolution.
CONIC SECTIONS

9. ELLIPSE
Ellipsesareformedwhentheplaneintersectsoneconeat
anangleotherthan90°.
CONIC SECTIONS

10. PARABOLA
Parabolasareformedwhentheplaneisparalleltothe
generatinglineofonecone.
CONIC SECTIONS

11. HYPERBOLA
Hyperbolasareformedwhentheplaneisparalleltothe
axisofrevolutionory-axis.
CONIC SECTIONS

12. DEGENERATE CONIC SECTIONS
Degenerate conic
sectionsareformed
whenaplaneintersects
theconeinsuchaway
thatitpassesthrough
theapex.

13. DEGENERATE CONIC SECTIONS
Degenerateconicsectionsareformedwhenaplane
intersectstheconeinsuchawaythatitpassesthrough
theapex.

14. INTERACTIVE 3D OF CONIC
SECTIONS
https://www.intmath.com/plane-analytic-
geometry/conic-sections-summary-interactive.php

15. COMMON PARTS OF THE CONIC SECTIONS
Theconicsectionsmayhavelookeddifferent;however,
theystillhavecommonparts.
VERTEX
CENTER
FOCUS
DIRECTRIX

16. COMMON PARTS OF THE CONIC SECTIONS
VERTEX
Vertexisanextremepointonaparabola,hyperbola,and
ellipse.Although,ellipsehasverticesandco-vertices.
with horizontal axis

17. COMMON PARTS OF THE CONIC SECTIONS
VERTEX
Vertexisanextremepointonaparabola,hyperbola,and
ellipse.Although,ellipsehasverticesandco-vertices.
with vertical axis

18. COMMON PARTS OF THE CONIC SECTIONS
FOCUS AND DIRECTRIX
Thefocusanddirectrixarethepointandthelineonaconicsection
thatareusedtodefineandconstructthecurverespectively.The
distanceofanypointonthecurvefromthefocustothedirectrixis
proportionasshownontheimagesbelowbythegreenlines.Ina
plane,thecirclehas no defineddirectrix.
with horizontal axis

19. COMMON PARTS OF THE CONIC SECTIONS
FOCUS AND DIRECTRIX
Thefocusanddirectrixarethepointandthelineonaconicsection
thatareusedtodefineandconstructthecurverespectively.The
distanceofanypointonthecurvefromthefocustothedirectrixis
proportionasshownontheimagesbelowbythegreenlines.Ina
plane,thecirclehas no defineddirectrix.
with vertical axis

20. COMMON PARTS OF THE CONIC SECTIONS
CENTER
Forcircles,thecenteristhepointequidistantfromany
pointonthesurface.

26. LookingattheCartesianplane,the
vertexisatthepoint(0,0)sinceitis
theextremepointoftheparabola.
The focusis at (3,0).
Inordertosolveforthedirectrix,
weneedtolookattheorientationof
theparabola.Sincetheformulahas
ahorizontalaxis,theformulaforthe
directrixwillbe�=−??????.Thus,the
equationofthedirectrixis�=−3.

28. LET’S TRY
GiventhecurveontheCartesianplanebelow,identifythe
focus,vertex,anddirectrix.

30. Lookingatthegraph,wecansee
thatthefociareat(−2−3)and(6,
−3).
Notethatthecenteristhemidpoint
ofthefoci.Thus,wehavethe
followingsolution:

32. LET’S TRY
Identifythecoordinatesofthefociandthecenterofthe
graphbelow.

34. Step1:Plotthevertexandthe
focus,andidentifythecurve.
LookingattheGatewayArch,it
lookslikeaparabola.
Step 2: Solve for the directrix.
Since the focus is at (0, −3),
the directrix is �= 3

43. ASSESSMENT

44. A.Identifytheconicsectionorthepartthatisbeing
described.
1.Thesearetheconicsectionsthatareformedwhenthe
planeintersectsthedouble-nappedconeinsuchawaythat
itpassesthroughtheapex.
2.Thisconicsectionisformedwhentheplaneisparallelto
theaxisofrevolution.
3.Itisthemidpointofthetwofociforellipseandhyperbola.
4.Itreferstotheextremepointofaparabola.
5.Thesearethecurvesthatareobtainedbetweenthe
intersectionofadouble-nappedconeandaplane.
ASSESSMENT

45. B.Usingthe image below, complete the table and
solve for the directrixgiven the vertices and foci.

48. Assignment:ChallengeYourself
Answerthefollowingquestions:
1.YousawAlbertplayingwithadouble-nappedconeandapaper.He
putthepaperontopofoneconeandsaidthathewasabletoforma
conicsection.Doyouagreewithhim?Explainyouranswer.
2.Aglasswasplacedonthetable.Ifyouholdaflashlightasshown
below,whatkindofcurvewillbeformedbyitsshadow?
3. If you shift a parabola with vertex at the origin, two units to the right,
what will be the new vertex?