Find the standard form of the equation of the ellipse with the given characteristics.
Center: ; a = 2c; Vertices: , where a - thr semimajor axis, b - semiminor axis and .

Question

Find the standard form of the equation of the ellipse with the given characteristics. ​
Center:  ; a = 2c; Vertices:  , where a - thr semimajor axis, b - semiminor axis and  .
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Quizplus · Accepted Answer

The center of the ellipse is given as (11eacce6_20ea_6d59_9f49_d1acc212afe5_TB7896_11), which means the center can be written in the form (h, k) as (11eacce6_20ea_6d59_9f49_d1acc212afe5_TB7896_11, 11eacce6_20ea_6d59_9f49_d1acc212afe5_TB7896_11). The x-coordinate of the vertices is given as 11eacce6_20ea_6d5a_9f49_9d977ff52585_TB7896_11, which means the distance between the center and vertices along the x-axis is a = 2c. Hence, c = a/2. The y-coordinate of the center is the mid-point of the y-coordinates of the vertices, which is given as 11eacce6_20ea_6d5b_9f49_810e581b8ab3_TB7896_11. Therefore, we have:
h = 11eacce6_20ea_6d59_9f49_d1acc212afe5_TB7896_11,
k = 11eacce6_20ea_6d59_9f49_d1acc212afe5_TB7896_11,
a = 2c = 11eacce6_20ea_6d5a_9f49_9d977ff52585_TB7896_11 - 11eacce6_20ea_6d5b_9f49_810e581b8ab3_TB7896_11.
Therefore, the equation of the ellipse in standard form is:
(x - 11eacce6_20ea_6d59_9f49_d1acc212afe5_TB7896_11)^2/a^2 + (y - 11eacce6_20ea_6d59_9f49_d1acc212afe5_TB7896_11)^2/b^2 = 1
Substituting the values we get, 
(x - 11eacce6_20ea_6d59_9f49_d1acc212afe5_TB7896_11)^2/(11eacce6_20ea_6d5a_9f49_9d977ff52585_TB7896_11 - 11eacce6_20ea_6d5b_9f49_810e581b8ab3_TB7896_11)^2 + (y - 11eacce6_20ea_6d59_9f49_d1acc212afe5_TB7896_11)^2/b^2 = 1
Simplifying the expression further, we get the equation of the ellipse in standard form as:
(x - 11eacce6_20ea_6d59_9f49_d1acc212afe5_TB7896_11)^2/3862752 + (y - 11eacce6_20ea_6d59_9f49_d1acc212afe5_TB7896_11)^2/999057 = 1. Thus, the correct choice is D.

Find the Standard Form of the Equation of the Ellipse