Question

Given the points (–3,k) and (2,0), for which values of k would the distance between the points be √34 ?A. 2 or -6B. 5 or -6C. 5 or 0D. 3 or -3

Answer 1

Step 1

State an expression for the distance between two points(D)

`D=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}`

Where

`\begin{gathered} x_2=2 \\ x_1=-3 \\ y_2=0 \\ y_1=k \\ D=\sqrt[]{34} \end{gathered}`

Step 2

Substitute and find k

`\begin{gathered} \sqrt[]{34}=\sqrt[]{(2-(-3))^2+(0-k)^2} \\ \sqrt[]{34}=\sqrt[]{(2+3)^5+(-k)^2} \end{gathered}`

`\begin{gathered} \sqrt[]{34}=\sqrt[]{5^2+k^2} \\ (\sqrt[]{34})^2=(\sqrt[]{25+k^2})^2 \\ 34=25+k^2 \\ k^2+25-34=0 \\ k^2-9=0 \\ \end{gathered}`

Factorizing using the difference of two squares gives

`\begin{gathered} (k-3)(k+3)=0 \\ k=3 \\ or \\ k=-3 \end{gathered}`

Hence option D is right