Contradiction in 2 different approaches for finding maximum and minimum values of $|{\vec{PA}}||{\vec{PB}}|$ The original question is :

Given three points on the xy plane, O(0,0) , A(1,0) , B(-1,0). Point P is moving on the plane satisfying the condition $(\vec{PA}\cdot\vec{PB})+3(\vec{OA}\cdot\vec{OB})=0$ . If the maximum and minimum values of $|{\vec{PA}}|| {\vec{PB}}|$ are M and m respectively then find the value(s) of $M^2+m^2$.

I found the solution to the exact problem $\color{red}{here.}$
But I approached the question in a different way.
My approach:
$$\vec{OA}=(1,0)$$
$$\vec{OB}=(-1,0)$$
Thus, $$\vec{OA}\cdot\vec{OB}=-1$$
Hence, according to the condition in the question: $$\vec{PA}\cdot\vec{PB}=3$$
Therefore, $$|\vec{PA}||\vec{PB}|\cos(\theta)=3$$
Or
$$|\vec{PA}||\vec{PB}|=\frac{3}{\cos(\theta)}$$
Where $\theta$ is the angle between vectors $\vec{PA}$ and $\vec{PB}$.
So according to what I've got, maximum value of $|\vec{PA}||\vec{PB}|$ occurs when $\theta=90°$ .
This contradicts the claim in the above given link that the maximum value of the required expression is 5.
So I want to know why is this contradiction arising,as I don't see any mistake in the answer given in the above link as well.
Thanks in advance.
 A: If you could construct two arbitrary vectors $v$ and $w$ of any lengths in any directions, then indeed there would be no maximum value of $\lVert v\rVert \lVert w\rVert$, because you could make $\theta$ as close to a right angle as you want and then make the vectors long enough so that $v\cdot w=3$.
But those are not the conditions under which you have to solve this problem.
If $\theta$ is near $90^\circ$ then $P$ is nearly on a circle whose diameter is the line segment $AB$. You cannot make the product $\lVert \vec{PA}\rVert \lVert \vec{PB}\rVert$ much larger than $2$. And of course $\vec{PA}\cdot\vec{PB}$ will be less than $3.$
Although your formula is true, you must honor the other constraints of the problem statement. That’s why you cannot maximize
$\lVert \vec{PA}\rVert \lVert \vec{PB}\rVert$
by setting $\theta$ to $90^\circ$ or close to $90^\circ$.
A: First you can not divide by zero!
Second your condition says AP and BP have to be perpendicular? how do you get this?
Your link does not work in my browser.
