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]:

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