<s>[INST] Minimize the ratio involving the ellipse 
Let $P$ be any point on the curve $\dfrac{x^2}{4}+\dfrac{y^2}{3}=1$,
and $A,B$ be two fixed points $\left(\dfrac{1}{2},0\right)$ and
$(1,1)$ respectively. Find the minimum value of
$\dfrac{|PA|^2}{|PB|}$.

Assume $x=2\cos\theta,y=\sqrt{3}\sin\theta$, then
$$\dfrac{|PA|^2}{|PB|}=\frac{\left(2\cos\theta-\frac{1}{2}\right)^2+(\sqrt{3}\sin\theta-0)^2}{\sqrt{\left(2\cos\theta-1\right)^2+(\sqrt{3}\sin\theta-1)^2}},$$
but which is not so easy to tackle.
WolframeAlpha gives the result $1$.
 [/INST] Denote $c = \cos \theta$ and $s = \sin\theta$.
We have
$$|\mathrm{PA}|^2 = (2c - 1/2)^2 + (s\sqrt 3 - 0)^2= c^2 - 2c + \frac{13}{4}$$
and
$$|\mathrm{PB}|^2 = (2c - 1)^2 + (s\sqrt 3 - 1)^2 
= c^2 - 4c + 5 - 2\sqrt 3\, s.$$
We have
\begin{align*}
 |\mathrm{PA}|^4 - |\mathrm{PB}|^2
 &= (c^2 - 2c + 13/4)^2 - (c^2 - 4c + 5 - 2\sqrt 3\, s)\\
 &= (c^2 - 2c + 13/4)^2 - c^2 + 4c - 5 + 2\sqrt 3\, s\\
 &\ge 0
\end{align*}
where we have used
$(c^2 - 2c + 13/4)^2 - c^2 + 4c - 5 > 0$ and
\begin{align*}
 &[(c^2 - 2c + 13/4)^2 - c^2 + 4c - 5]^2 - (2\sqrt 3\, s)^2\\
 =\,& [(c^2 - 2c + 13/4)^2 - c^2 + 4c - 5]^2 - 12(1-c^2)\\
 =\,&\frac{1}{256}(64c^6 - 448c^5 + 1776c^4 - 4128c^3 + 6524c^2 - 6236c + 4849)(2c-1)^2\\ 
 \ge\,& 0.
\end{align*}
On the other hand, when $\theta = \frac{5\pi}{3}$,
we have $|\mathrm{PA}|^4 - |\mathrm{PB}|^2 = 0$ and $\frac{|\mathrm{PA}|^2}{|\mathrm{PB}|} = 1$.
Thus, the minimum of $\frac{|\mathrm{PA}|^2}{|\mathrm{PB}|}$ is $1$.
</s>