I've done the non-bold part. Help me with theBy introducing suitable coordinate axes, show that a curve with the above “constant sum” property indeed has an equation of the form

$\displaystyle \displaystyle{ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 }$(It is a good idea to start with the two square root terms, representing the distances F1 P and F2P, on opposite sides of the equation.)

Show also thatanyequation of this form is obtainable by suitable choice of $\displaystyle F_1$,$\displaystyle F_2$and$\displaystyle F_1P+F_2P$.bold part, i.e. thekeyword.any