Two ways to define an ellipse

Question

I have some problems in understanding this problem, because I'm stuck in some purely mathematical definitions and do not know how to proceed, appreciate to some that I can say which is the best way forward to solve this problem, the problem is as follows:

Given two conditions to define a ellipse:

A. By the condition that the distances $a'$, $a''$ between any two points on the ellipse and the two focal points add to a constant $$ a'\text{+}a''\text{=}2a $$

given by twice the length a of the long axis and,

B. by the equation

$$ 1 = \frac{x^{2}}{a^{2}} +\frac{y^{2}}{b^{2}}$$

in Cartesian coordinates with the $x$ and $$-axis coinciding with the long and the short axis of the ellipse respectively.

1. Show $A$ is a sufficient condition for $B$.
2. Show that $A$ is also a necessary condition for $B$.

I appreciate your comments and insights.

Answer 1

This is not a complete solution but a guide to solve your problem.

1. To prove the sufficient part, start with first equation $A$ to derive the second equation $B$: by taking a point $(x,y)$ on ellipse and writing $a'$ and $a"$ in terms of $x$ and $y$, we get

$\sqrt{(x+c)^2 + y^2} + \sqrt{(x-c)^2 + y^2} = 2a$

where $c$ = distance of a focus from the origin.

Simplify this equation using $c^2 = a^2 - b^2$ to get the equation $B$.

2. For the necessary part, you do the reverse of the above, i.e., start with the equation $B$ to derive the equation $A$: by writing $a'$ and $a"$ in L.H.S of the equation $B$ in terms of $x$ and $y$ and simplify it using the equation $A$ to get the R.H.S, i.e., $2a$.

Hope this helps.