<s>[INST] Can I find the equation of an ellipse with these points? How can I solve an ellipse with its major axis on the $x$-axis, given one focus, and two points on the ellipse, one of which I know to be on the major axis?
 [/INST] Let $F$, $V$, and $P$ be the given focus, the given vertex (point on the major axis), and the other point. We'll assume that $V$ is the "closer" vertex to $F$ (that is, $|VF| = a-c$, where $a$ is the semi-major axis, and $c > 0$ is the focal radius; otherwise, $|VF|=a+c$, and the argument proceeds similarly).
The (fraction-free) standard form of the ellipse is
$$b^2 ( x - h )^2 + a^2 ( y - k )^2 = a^2 b^2$$
where $(h,k)$ is the center, $a>0$ is the semi-major axis, and $b>0$ is the semi-minor axis. With the major axis given to be on the $x$-axis, we have $k=0$. Let us assume that the focus $F$ is positioned at the origin, and that the ellipse's center is to the left of that focus, so that $h = -c < 0$. The equation becomes
 $$b^2 ( x + c )^2 + a^2 y^2 = a^2 b^2$$
We therefore need to determine the three quantities $a$, $b$, $c$.
Let us position focus $F$ at the origin, and assign coordinates to the other points thusly: $V(v,0)$ and $P(p,q)$ (with $v > 0$ and $v \ge p$). We have three equations:
$$\begin{align}
v &= a - c &(1) \\
a^2 &= b^2 + c^2 &(2) \\
b^2 ( p + c )^2 + a^2 q^2 &= a^2 b^2 &(3)
\end{align}$$
We use $(1)$ to eliminate $c$ from $(2)$ and $(3)$:
$$\begin{align}
b^2 = a^2 - c^2 = ( a - c )( a + c ) &= v ( 2a - v ) &(4) \\
b^2(p+a-v)^2+a^2q^2 &= a^2 b^2 &(5)
\end{align}$$
... and then use $(4)$ to eliminate $b^2$ from $(5)$:
$$v(2a-v)(p+a-v)^2+a^2q^2 = a^2 v(2a-v)$$
whence
$$a^2 \left(p^2 + q^2- (2v-p)^2\right) + 2 a v\left(v-p\right)\left(2v-p\right) - v^2 \left(v-p\right)^2 = 0$$
so that
$$a = \frac{v \left(v-p\right) \left(-\left( 2v - p \right) \pm \sqrt{p^2+q^2}\right)}{p^2 + q^2- (2v-p)^2} = \frac{v\left(v-p\right)}{2v-p \pm \sqrt{p^2+q^2}}$$
with the "$\pm$" chosen to make $a > v > 0$, and 
$$b = \sqrt{\;v(2a-v)\;} \qquad\qquad c = a - v$$
(You should double-check my algebra.)
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