Finding b in terms of a and c. Regarding an Ellipse

Question

If I have an ellipse with foci at $(c,0)$and $(-c,0)$, and passes through the point $(0,b)$ where $b>0$. Find $b$ in terms of $a$ and $c$. Where $a$ is the distance from the center to the end of the major axis.

Answer 1

As you know, for every point P on an ellipse with foci at C and C' it follows: $dPC+dPC'=k$, for a real constant $k$. (Note: $dPC$ represents the distance between points $P$ and $C$)

In this case we have: $C(c,0)$ and $C'(-c,0)$. Also: $B(0,b)$ is on the described ellipse. Let $AA'$ be the mayor axis, where $A(a,0)$ and $A'(-a,0)$. Then,

$$dAC+dAC'=(a-c)+(a+c)=2a=k$$

Therefore, $\tag{1} k=2a$

By the Pythagorean theorem:

$$dBC=dBC'=\sqrt{c^2+b^2}$$

By the definition of ellipse: $dBC+dBC'=2\sqrt{c^2+b^2}=k$. By (1), we have:

$$2\sqrt{c^2+b^2}=2a \Rightarrow c^2+b^2=a^2 \Rightarrow b=\sqrt{a^2-c^2}$$