Parabola
Have You Ever Heard About Quadratic Equation?
Obviously....
It Is A Second Degree Equation,Which Can Be Solved By Either Factarization Or Completing Square.Quadratic Equation Have Always 2 Roots(Values Of Unknown Variable).
But We Have To Discuss Parbola!!!
Why We Talking About Quadratic Equation???
Actually The Graph Of Quadratic Equation Is Parabola.
So How We Define A Parabola Or What Actually Is A Parabola?
Definition#1
"The Locus Of A Point Moving In A Plane Such That The Distance From A Fixed Point (focus) And A Fixed Line Builds Up A Ratio Which Is Equal To 1"
Definition#2
"Conic Section Having Eccentricity Equals To 1"
Eccentricity e=c/a
where "c" is the distance from focus to the center of Parabola
"a" is the directrix of parabola
Explanation
Lets Explain Parabola....
Points To Be Note:
•Parbola Can Be Lie Anywhere In The Plane.
•Parabola Is An Open Curve.
•Parabola Is A Infinite Curve.
Equation Of A Parabola
For Writing Equation Of A Parabola We Have 2 Cases:
1)When Center Lies On Origin(0,0).
2)When Center Lies On A Point Other Than Center.
Case#1
First We Discuss When The Center Of Parabola Lies On Origin:
The Equation Is Given Below
y=4px
Where X And And Y Are Variables And P Is A Constant.
P Is The Distance Between Focus And Vertex.
The Above Equation Can Also Be Write In The Terms Of y
x=4py
Replacing X And Y Change The Orientation Of Parabola In Plane.
Case#2
Now We Explain The Parabola When Center Lies Any Where In Plane Other Than Origin:
For This Equation Is Given Below
(y-k)²=4p(x-h)
Where h And k Are The Coordinates Of Center.
Terminologies Related To Parabola
Now We Explain Some Terminologies Related To A Parabola.
Focus
It Is A Fixed Point From Which The Distance Is Taken On A Parabola.It Lies On Axis Of Symmetry Of Parabola.
It Is Generally Denoted By "f".
Axis Of Symmetry
It Is Line Segment Which Divide A Parabola Into 2 Equal Parts.It Is Normal To The Directrix And Pass Through Focus And Vertex.It Is Generally Denoted By A Simple Linear Equation.
Directrix
It Is Also A Line Line Segment Like Axis Of Symmetry But It Is Normal To The Axis Of Symmetry.It Lies In Closed Side Of Parabola.It Is Denoted By A Linear Equation.
Relationship Between Focus,Vertex And Directrix
Focus, Vertex And Directrix Are Related To Each Other By In This Way:
If
"f" Is Focus
"v" is Vertex
"m" is point Of Intersection Of Vertex And Directrix,then
distance between f and v=distance between v and m
|fv|=|vm|
.
Focal Chord
Any Line Segment Having End Points On Parabola And Pass Through Focus.
Latus Rectum
Any Line Segment Parallel To Directrix Having Points On Parabola And Pass Through Focus.
Relationship Between Focal Chord And Latus Rectum
Focal Chord And Latus Rectum Are Related To Each Other As:
Every Latus Rectum Is A Focal Chord.
Every Focal Chord Is Not A Latus Rectum.
Orientations Of A Parabola
The Orientation Of A Parabola Depends On Two Things
1)Sign Of "p" in Equation Of Parabola
2) Leading And Defined Variables
1)Sign Of "p"
p is the distance between focus and vertex
if p is positive (+) then parabola will either open rightward or upward.
if p is negitive (-) then parabola will either open leftward or downward.
2) Leading And Defined Variables
If y is leading variable (variable to which equation Equals) Then Parabola Is Open Leftward Or Rightward
If x Is Leading Variable Then Parabola Is Open Upward Or Downward.
Values Of Terms When Equation Is
When Equation Is x²=4py Then
Focus (0,p)
Directrix y=-p
Vertex(0,0)
When Equation Is y²=4px Then
Focus (p,0)
Directrix y=-p
Vertex(0,0)
When Equation Is (x-h)²=4p(y-k) Then
Focus (h,p+k)
Directrix y=k-p
Vertex(h,k)
When Equation Is (y-k)²=4p(x-h) Then
Focus (h+p,k)
Directrix x=h-p
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