First line: Two polynomials.

pls = {(b^2 + a^2 k^2) x^2 + 2 a^2 k m x + (-a^2 b^2 + a^2 m^2), 
       (b^2 - a^2 k^2) y^2 - 2 b^2 m y + (-a^2 b^2 k^2 + b^2 m^2)}

Second line: The quadratic coefficients(x^2 and y^2) of the first-line polynomials, respectively.

coe = {b^2 + a^2 k^2, b^2 - a^2 k^2}

Third line: The two polynomials are each divided by their own quadratic coefficient, yielding a new set of polynomials.

pls / coe

The output is in the following form:The output doesn't match the desired form.

{(-a^2 b^2 + a^2 m^2 + 2 a^2 k m x + (b^2 + a^2 k^2) x^2)/(
 b^2 + a^2 k^2), (-a^2 b^2 k^2 + b^2 m^2 - 
  2 b^2 m y + (b^2 - a^2 k^2) y^2)/(b^2 - a^2 k^2)}

The desired output should look like this:

Order the polynomials by decreasing degree in x or y, with separate groupings for the quadratic, linear and constant terms, joined by plus or minus signs. as followings:

{x^2 + (2 a^2 k m x)/(b^2 + a^2 k^2) + (-a^2 b^2 + a^2 m^2)/(
  b^2 + a^2 k^2), 
 y^2 + (-2 b^2 m y)/(b^2 - a^2 k^2) + (-a^2 b^2 k^2 + b^2 m^2)/(
  b^2 - a^2 k^2)}

I made these attempts but didn't get the wanted result form. How should I proceed to obtain the target form mentioned above?

result = {Expand[pls[[1]]/coe[[1]]], Expand[pls[[2]]/coe[[2]]]}