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Question 21661: I have an ellipse question, and I either have drawn a blank or there's a mistake somewhere. I know that I'm supposed to factor out where I can, but I just can't seem to make it work. I need to find the foci, vertex and center using the equation: 9x^2 + y^2 + 18x + 4y + 4 = 0. Can anyone help me????
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! 9x^2 + y^2 + 18x + 4y + 4 = 0. STANDARD FORM FOR EQN.OF ELLIPSE IS
{(X-H)^2/A^2}+{(Y-K)^2/B^2}=1
LET US TRY TO PUT THE GIVEN EQN.IN THIS FORM
{9(X^2+2X+1-1)}+{Y^2+4Y+4-4}=-4
9(X+1)^2+(Y+2)^2=-4+4+9=9
DIVIDING WITH 9 THROUGH OUT WE GET
[(X+1)^2/1]+[(Y+2)^2/9]=1...THIS IS IN THE STANDARD FORM WITH
H=-1...K=-2.....A=1.....B=3.....SINCE WE HAVE A AS LESS THAN B WE GET
CENTRE =(H,K)=(-1,-2)
ECCENTRICITY =E =SQ.RT{(B^2-A^2)/B^2}=SQ.RT.{(9-1)/9}=2SQ.RT(2)/3
FOCI ARE GIVEN BY (H,K+BE) AND (H,K-BE)=(-1,-2+3*2SQ.RT2/3)
AND(-1,-2+3*2SQ.RT2/3)
THAT IS
=(-1,-2+2SQ.RT2)
AND(-1,-2+2SQ.RT2)
THERE IS NO VERTEX FOR ELLIPSE.
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