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

Apr 2, 2018

${x}^{2} / {12}^{2} + {y}^{2} / {\left(\sqrt{119}\right)}^{2} = 1$

#### Explanation:

The foci at $\left(- 5 , 0\right) \mathmr{and} \left(5 , 0\right)$ tells us that the major axis is horizontally oriented, therefore, the general form is:

${\left(x - h\right)}^{2} / {a}^{2} + {\left(y - k\right)}^{2} / {b}^{2} = 1 , a > b \text{ [1]}$

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

$k = 0 \text{ [2]}$
$h - \sqrt{{a}^{2} - {b}^{2}} = - 5 \text{ [3]}$
$h + \sqrt{{a}^{2} - {b}^{2}} = 5 \text{ [4]}$

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

$2 h = 0$

$h = 0 \text{ [5]}$

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

${\left(x\right)}^{2} / {a}^{2} + {\left(y\right)}^{2} / {b}^{2} = 1 \text{ [1.1]}$

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

$a = \frac{24}{2}$

$a = 12 \text{ [6]}$:

Substitute equation [6] into equation [1.1]:

${\left(x\right)}^{2} / {12}^{2} + {\left(y\right)}^{2} / {b}^{2} = 1 \text{ [1.2]}$

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

$0 + \sqrt{{12}^{2} - {b}^{2}} = 5$

$- {b}^{2} = 25 - 144$

$b = \sqrt{119} \text{ [7]}$

Substitute equation [7] into equation [1.2]:

${x}^{2} / {12}^{2} + {y}^{2} / {\left(\sqrt{119}\right)}^{2} = 1$