SOLUTION: How do mathematicians get the formula: lal = 1/4c (absolute value of 'a' equals one-fourth times 'c')? This formula is used to determine the equation of a parabola. Please explain

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Question 36142: How do mathematicians get the formula: lal = 1/4c (absolute value of 'a' equals one-fourth times 'c')? This formula is used to determine the equation of a parabola. Please explain how mathematicians got this formula. Thank you for your time.
Julia
Answer by venugopalramana(3286) (Show Source):
You can put this solution on YOUR website!
How do mathematicians get the formula: lal = 1/4c (absolute value of 'a' equals one-fourth times 'c')? This formula is used to determine the equation of a parabola. Please explain how mathematicians got this formula. Thank you for your time.
Julia
I TAKE IT THAT YOU ARE REFERRING TO EQN. OF PARABOLA..IN STD. FORM
Y^2=4AX...........WITH AN EQN.IN THE FORM .......Y^2=CX...OR...SAY...Y^=X..OR SO...WELL

PARABOLA IS THE LOCUS OF A POINT (P SAY ..WITH COOORDINATES X AND Y )WHICH MOVES SUCH THAT ITS DISTANCE FROM A POINT CALLED FOCUS (F SAY) = ITS DISTANCE FROM A LINE CALLED DIRECTRIX.
IN THE STANDARD FORM , A PERPENDICULAR FROM FOCUS ( F G SAY)TO DIRECTRIX IS TAKEN AS X AXIS.
MID POINT ON THIS IS TAKEN AS THE ORIGIN O (0,0).AND A LINE PARALLEL TO DIRECTRIX THROUGH O AS THE Y AXIS.
LET OF = A ....SINCE OF=OG BY CONSTRUCTION,AS O IS THE MIDPOINT OF FG, WE HAVE OF=OG=A.
HENCE AS PER THE DEFINITION OF PARABOLA ,O IS A POINT SUCH THAT ITS DISTANCE FROM FOCUS F =OF IS AME AS ITS DISTANCE FROM DIRECTRIX =OG.
HENCE O LIES ON THE PARABOLA.IT IS CALLED VERTEX.
HENCE COORDINATES OF F..FOCUS ARE (A,0)
EQN. OF DIRECTRIX IS X=-A..SINCE IT IS PARALLEL TO Y AXIS AND IS AT A CONSTANT DISTANCE OF -A FROM IT.
NOW IF P IS ANY POINT ON THE PARABOLA , ITS DISTANCE FROM F IS GIVEN BY

D^2=(X-A)^2+(Y-0)^2
THIS IS EQUAL TO ITS DISTANCE FROM DIRECTRIX = X+A
HENCE WE HAVE
(X+A)^2=(X-A)^2+Y^2
Y^2=X^2+A^2+2AX-(X^2+A^2-2AX)=4AX
HENCE Y^2 =4AX IS THE EQN.OF PARABOLA IN ITS STANDARD FORM
NOW IN A GENERAL PROBLEM IF IT IS GIVEN THAT
Y^2=8X ..0R..IN GENERAL....CX...THEN IT MEANS
Y^2=4*2X...OR...A=2.....IN GENERAL .....Y^2=4*(C/4)*(X)...HENCE COMPARING WE GET
A=C/4
HOPE YOU UNDERSTOOD.