Give standard equation for the ellipse with the given characteristics: Major axis of length 24 Foci:(plus/minus 5,0)?

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Answer 1 · 2018-04-02T01:11:00

#x^2/12^2+y^2/(sqrt119)^2=1#

The foci at #(-5,0) and (5,0)# tells us that the major axis is horizontally oriented, therefore, the general form is:

#(x-h)^2/a^2+(y-k)^2/b^2=1, a > b" [1]"#

The location of the foci allows us to write the following equations:

#k = 0" [2]"#
#h-sqrt(a^2-b^2) = -5" [3]"#
#h + sqrt(a^2-b^2) = 5" [4]"#

Add equation [3] to equation [4] and solve for the value of h:

#2h = 0#

#h = 0" [5]"#

Substitute equations [2] and [5] into equation [1]:

#(x)^2/a^2+(y)^2/b^2=1" [1.1]"#

The major axis length of #24# allows us to find the value of a:

#a = 24/2#

#a = 12" [6]"#:

Substitute equation [6] into equation [1.1]:

#(x)^2/12^2+(y)^2/b^2=1" [1.2]"#

Substitute equations [5] and [6] into equation [4] and solve for the value of b:

#0 + sqrt(12^2-b^2) = 5#

#-b^2 = 25-144#

#b = sqrt119" [7]"#

Substitute equation [7] into equation [1.2]:

#x^2/12^2+y^2/(sqrt119)^2=1#