If P = (-3, 1) and Q = (x, 4), find all numbers x such that the vector represented by line PQ has length 5.
1. Calculate the vector $\displaystyle \overrightarrow{PQ} = \overrightarrow{OQ} - \overrightarrow{OP} = ((x-(-3)), 3)$
2. Calculate the magnitude of the vector $\displaystyle \overrightarrow{PQ}$ which has to be 5:
$\displaystyle \sqrt{(x+3)^2+3^2} = 5$ Square both sides:
$\displaystyle (x^2+6x+9) +9 = 25~\implies~x^2+6x-7=0~\implies~x=-7~\vee~x=1$
3. Therefore $\displaystyle Q_1 (-7,4)$ or $\displaystyle Q_2(1,4)$
Note that $\displaystyle Q\in\mathcal{C}$, where $\displaystyle \mathcal{C}$ is the circle centered at $\displaystyle P$ with the radius $\displaystyle r=5$.
Just write the equation of $\displaystyle \mathcal{C}$ and substitute $\displaystyle Q$ to this equation and solve $\displaystyle x$ as earboth done.