Question

given the points (2,k) and (0,-6) for whick values of k would the distance between points sqaure root of 5

Answer 1

Given the points (2, k) and (0, -6), and the distance is √5

The distance between two points is calculated by the formula:

`\begin{gathered} d=\sqrt[]{(y_2-y_1)^2+(x_2-x_1)^2} \\ \text{where,} \\ (x_1,y_1)=(2,k) \\ (x_2,y_2)=(0,-6) \\ d=√(5) \end{gathered}`

Hence,

`\begin{gathered} \sqrt[]{5}=\sqrt[]{(-6-k)^2+(0-2)^2} \\ \sqrt[]{5}=\sqrt[]{(-6-k)^2+(-2)^2} \\ \sqrt[]{5}=\sqrt[]{(-6-k)^2+4} \\ \text{squaring both sides} \\ 5=(-6-k)^2+4 \\ (-6-k)(-6-k)+4=5 \\ 36+6k+6k+k^2+4=5 \\ k^2+12k+36+4-5=0 \\ k^2+12k+35=0 \\ \text{factorize completely,} \\ (k^2+5k)+(7k+35)=0 \\ k(k+5)+7(k+5)=0 \\ (k+7)(k+5)=0 \\ k+7=0 \\ k=-7 \\ k+5=0 \\ k=-5 \end{gathered}`

Therefore, the values of k would be -5 or -7 [option A is correct]