Q.Find the equation and eccentricity of the ellipse if:

Setting $0<e<1$ in the conic definition gives the ellipse — its distance from the focus is less than $e$ times its distance from the directrix.

Standard form: $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, where $b^2=a^2(1-e^2)$ (so $b<a$). Foci $(\pm c,0)$ with $c^2=a^2-b^2$ (equivalently $c=ae$); vertices $(\pm a,0)$; directrices $x=\pm a/e=\pm a^2/c$. Major axis $2a$ (through the foci), minor axis $2b$; latus rectum $=2b^2/a$.

Theorem 5.5 (defining sum property). For any point $P$ on the ellipse, $SP+S'P=2a$ (constant) — the "string and two pins" construction.

Centre $(h,k)$: major axis parallel to $x$: $\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1$, vertices $(h\pm a,k)$, foci $(h\pm c,k)$. Major axis parallel to $y$: swap the roles of the two denominators — vertices $(h,k\pm a)$, foci $(h,k\pm c)$. The larger denominator always marks the major-axis direction.

Remarks. As $e\to0$, the ellipse rounds into a circle ($e=0$ is the circle's own eccentricity, directrix at infinity). Auxiliary circle $x^2+y^2=a^2$ (major axis as diameter) parametrises the ellipse (§Conic Parametric Forms); incircle $x^2+y^2=b^2$ (minor axis as diameter). …