Find the standard form of the equation of the ellipse with the given characteristics?

Foci: (0, 0) and (4, 0)
Major Axis of length 8

2 Answers
Shwetank Mauria
Jul 4, 2018

Equation is #(x-2)^2/16+y^2/12=1#

Explanation:

As focii are #(0,0)# and #(4,0)#, center of ellipse is midpoint i.e. #(2,0)# and major axis is #8#, equation is of the form

#(x-2)^2/4^2+(y-0)^2/b^2=1#

where #b# is half minor axis.

As distance between focii is #4# and major axis is #8#, eccentricity is #4/8=1/2# and

#(1/2)^2=1-b^2/4^2#

or #b^2/16=1-1/4=3/4#

and #b^2=12#

Hence equation of ellipse is
#(x-2)^2/16+(y-0)^2/12=1#

or #(x-2)^2/16+y^2/12=1#

Shwetank Mauria
Jul 4, 2018

Equation is #(x-2)^2/16+y^2/12=1#

Explanation:

As focii are #(0,0)# and #(4,0)#, center of ellipse is midpoint i.e. #(2,0)# and major axis is #8#, equation is of the form

#(x-2)^2/4^2+(y-0)^2/b^2=1#

where #b# is half minor axis.

As distance between focii is #4# and major axis is #8#, eccentricity is #4/8=1/2# and

#(1/2)^2=1-b^2/4^2#

or #b^2/16=1-1/4=3/4#

and #b^2=12#

Hence equation of ellipse is
#(x-2)^2/16+(y-0)^2/12=1#

or #(x-2)^2/16+y^2/12=1#

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