Q.Find the equation of the ellipse with semi-major axis $6$, eccentricity $\dfrac{2}{3}$, and major axis along the $x$-axis.

Using the focal-distance definition of the ellipse and the point at the end of the minor axis, $a^2=b^2+c^2$ with $c>0$ forces $a>b$.

Let the ellipse have foci $S(c,0)$ and $S'(-c,0)$ (with $c>0$, since a genuine ellipse — not a circle — has two distinct foci), and let the constant sum of focal distances be $2a$ (this defines the semi-major axis length $a$, so major axis $=2a$).

Let $B=(0,b)$ be the point where the ellipse meets the $y$-axis (the end of what will be called the minor axis, length $2b$). By symmetry about the $y$-axis, $B$ is equidistant from both foci:

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

By the defining property of the ellipse, $BS+BS'=2a$, so:

$2\sqrt{b^2+c^2}=2a\implies\sqrt{b^2+c^2}=a\implies a^2=b^2+c^2.$

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