the proof of the eccentricity of an ellipse

Question

independent from the directrix, the eccentricity is defined as follows:

For a given ellipse:

• the length of the semi-major axis = $a$

• the length of the semi-minor = $b$

• the distance between the foci = $2c$

• the eccentricity is defined to be $\dfrac{c}{a}$

now the relation for eccenricity value in my textbook is $\sqrt{1- \dfrac{b^{2}}{a^{2}}}$

which I cannot prove.

Answer 1

For two focus $A,B$ and a point $M$ on the ellipse we have the relation $MA+MB=cst$.

We can evaluate the constant at $2$ points of interest :

• on the intersection of major axis and ellipse closest to $A$

$MA+MB=2MA+AB=2(a-c)+2c=2a$

• on an intersection of minor axis and ellipse

we have $MA=MB$ and by pythagore $MA^2=c^2+b^2$
Combining all this gives $4a^2=(MA+MB)^2=(2MA)^2=4MA^2=4c^2+4b^2$
$\implies a^2=b^2+c^2$

Please try to solve by yourself before revealing the solution.