Question 1048583


Find an equation for the ellipse that satisfies the given conditions:
Foci (1, ±4), 
vertices (1, ±7) 


The center is midway between the two foci, so ({{{h}}},{{{ k}}}) = ({{{1}}}, {{{0}}}),  by the Midpoint Formula. 
Each focus is {{{4}}} units from the center, so {{{c = 4}}}. 

The vertices are {{{7}}} units from the center, so {{{a = 7}}}. 
Since the focus and vertex are above and below each other, rather than side by side, I know that this ellipse must be taller than it is wide. Then {{{a^2}}} will go with the {{{y}}}  part of the equation.

{{{(y-k)^2/a^2+(x-h)^2/b^2 = 1}}}

The equation {{{b^2 = a^2 - c^2}}} gives me 
{{{b^2 = 7^2 - 4^2}}}, 
{{{b^2 = 49 - 16}}}
{{{b^2 = 33}}}
and this is all I need to create my equation:

   {{{ (y - 0)^2 / 49 + (x-1)^2 / 33= 1}}}

    {{{y ^2 / 49 + (x-1)^2 / 33= 1}}}