Question

If the distance between the points A(2, −2) and B(−1, x) is 5, then

(a) x = −3 or x = 4
(b) x = 3 or x = −4
(c) x = −6 or x = 2
(d) x = 6 or x = −2

Answer 1

(c) x = −6 or x = 2
The given points are A(2, −2) and B(−1, x) and AB = 5.
Then, $\left(x_1=2,y_1=-2\right)\text{and}\left(x_2=-1,y_2=x\right)$
Therefore,
$$\begin{gathered} AB=5 \\ \Rightarrow \sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}=5 \\ \Rightarrow \sqrt{{\left(-1-2\right)}^2+{\left(x+2\right)}^2}=5 \\ \Rightarrow {\left(-3\right)}^2+{\left(x+2\right)}^2=25 \\ \Rightarrow 9+x^2+4x+4=25 \\ \Rightarrow x^2+4x+13=25 \\ \Rightarrow x^2+4x-12=0 \\ \Rightarrow \left(x+6\right)\left(x-2\right)=0 \\ \Rightarrow x=-6\text{or}x=2 \end{gathered}$$