Question

If the distance between the points A(4, p) and B(1, 0) is 5, then

(a) p = 4 only
(b) p = −4 only
(c) p = ±4 only
(d) p = 0

Answer 1

(c) p = ±4 only
The given points are A(4, p) and B(1, 0) and AB = 5.
Then, $\left(x_1=4,y_1=p\right)\text{and}\left(x_2=1,y_2=0\right)$
Therefore,
$$\begin{gathered} AB=5 \\ \Rightarrow \sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}=5 \\ \Rightarrow \sqrt{{\left(1-4\right)}^2+{\left(0-p\right)}^2}=5 \\ \Rightarrow {\left(-3\right)}^2+{\left(-p\right)}^2=25 \\ \Rightarrow 9+p^2=25 \\ \Rightarrow p^2=16 \\ \Rightarrow p=\pm \sqrt{16} \\ \Rightarrow p=\pm 4 \end{gathered}$$