Question 1186200
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Find the standard form of the equation of the ellipse with the given characteristics.
Center: (3, −8); a = 8c; foci: (3, −9), (3, −7)
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Notice that the foci lie on vertical line x = 3.


It means that the major axis of the ellipse is vertical  x= 3, parallel to y-axis,

and the minor axis is horizontal y= -8, parallel to x-axis.


The distance between the foci is  2c = -7 - (-9) = 2;  hence, the linear eccentricity is half of this distance c = 2/2 = 1.


Further, the major semi-axis  a = 8c = 8;  hence, the minor semi-axis is  b = {{{sqrt(a^2-c^2)}}} = {{{sqrt(8^2-1^2)}}} = {{{sqrt(63)}}}.


Now we are ready to write the standard form equation of the ellipse


    {{{(x-3)^2/b^2}}} + {{{(y+8)^2/a^2}}} = 1,

or

    {{{(x-3)^2/63}}} + {{{(y+8)^2/64}}} = 1.
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Solved.