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Question 32653: Can someone explain to me how to graph (x+2)²/9+(y-3)²/4=1 and other ellipses and how to find the equation of an ellipse from its graph?
Found 2 solutions by mukhopadhyay, venugopalramana:
Answer by mukhopadhyay(490)   (Show Source):
You can put this solution on YOUR website! You will need to find the critical co-ordinates (on a plane) of an ellipse to draw graph.
The critical points (co-ordinnates) of an ellipse are: Vertices and end points of its minor axis;
(x+2)²/9+(y-3)²/4=1;
The center of the above ellipse is at (-2, 3); the major axis is y=3 (parallel to x-axis) and the minor axis is x=-2 (parallel to y-axis)
The vertices of this ellipse are (-2-3,3) and (-2+3,3); addition of 3 to or subtraction of 3 from the x-coordinate of the center comes from sqrt(9)
when simplified, they are (-5,3) and (1,3);
The end points of the minor axis are (-2, 3+2) and (-2, 3-2); addition of 2 to or subtraction from of 2 from the y-coordinate of the center comes from sqrt(4)
when simplified, they are (-2,5) and (-2,1);
Plot these 4 co-ordinates on the graph paper and join them to form an oval shaped diagram.
Answer by venugopalramana(3286)  (Show Source):
You can put this solution on YOUR website! EQN.OF ELLIPSE IS ….(X+2)^2/9 +(Y-3)^2/4 = 1………….I
STD.EQN.OF ELLIPSE IS
(X-H)^2/A^2 + (Y-K)^2/B^2=1…………………..II………….COMPARING WITH ABOVE EQN.I
WHERE H,K IS CENTRE...SO H=-2 AND K=3 HENCE CENTRE IS (-2,3)..
A^2=9….OR……A=-3…. AND…….A =+3
B^2=4….OR……B=-2….AND….B=+2
MINOR AXIS =2B=4…
MAJOR AXIS =2A=6
NOW TO PLOT THE ELLIPSE YOU NEED
1.CENTRE COORDINATES…HERE THEY ARE (-2,3)
2.LENGTHS OF AXES….HERE…..6 AND 4.
3.EQNS.OF AXES…..MINOR AXIS IS…….X=H…OR…X=-2….OR…X+2=0
MAJOR AXIS IS …….Y=K…..OR……Y=3….OR……..Y-3=0
HAVING FOUND THE REQUIRED DATA,TAKE A SUITABLE SCALE…SAY 1 CM=1 UNIT OR SO AND PLOT
1.MARK ORIGIN…O,X AXIS AND Y AXIS
2.PLOT CENTRE (-2,3)…CALL IT C
3.DRAW MAJOR AXIS Y-3=0…THIS IS A LINE THROUGH CENTRE PARALLEL TO X AXIS
4.DRAW MINOR AXIS X+2=0….THIS IS A LINE THROUGH THE CENTRE PARALLEL TO Y AXIS.
5.MARK MAJOR AXIS LENGTH OF 6 ALONG THE MAJOR AXIS PLOTTED ABOVE….TAKING 3 UNITS ON EITHER SIDE OF CENTRE.CALL IT AA'
6.MARK MINOR AXIS LENGTH OF 4 ALONG THE MINOR AXIS PLOTTED ABOVE…TAKING 2 UNITS ON EITHER SIDE OF CENTRE.CALL IT BB'
7.ELLIPSE PASSES THROUGH A,A',B AND B'.JOIN THEM SMOOTHLY BY A SMOOTH ELLIPTICAL CURVE.
8.IF YOU NEED YOU CAN PLOT ADDITIONAL POINTS BY GIVING ARBITRARY VALUES FOR X AND FINDING CORRESPONDING VALUE OF Y
THE SAME PROCEDURE YOU CAN REVERSE TO FIND THE EQN.OF AN ELLIPSE FROM THE GRAPH
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