Iam can i solve this? show your work!

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Answer is

   
   
    foci at  and 



the ellipse has a horizontal radius of 7 and a vertical radius of 6. the major axis is horizontal.

standard form for an ellipse with a horiz. major axis is

   

the foci are at (±c, 0) and c² = a² - b².

the vertices are at (±a, 0).
for this ellipse in this question, vertices are at (±7, 0), so a = 7

the co-vertices are at (0, ±b).
for this ellipse in this question, co-vertices are at  (0, ±6), so b = 6

so the equation for this ellipse is

   

find foci. since the foci are at (±c, 0) and c² = a² - b², we solve for c

   

therefore, the foci are at 

    and 



an alternate way to find the equation: you can figure out equation of the ellipse using transformations of a circle.

a circle with a radius of 1 has this equation:

   

consider y = x². if we want to horizontally expand this parabola by a factor of 2, we need to replace x with (x  · 1/2) or x/2, aka multiply x by the reciprocal of the scale factor.

if we want to horizontally expand this circle by a factor of a, we need to replace x with x/a. 

   

since the radius is one, the horizontal radius will be stretched out to a units.
 
vertically expanding it by a factor of b, we replace y with y/b

   

so we can start with a circle. then with that circle, horizontally expand it by a factor of 7 units and vertically expand it by a factor of 6 units.