Q39 / 89 · KVPY 2017 (5 Nov SB/SX) · Mathematics
Let $\mathrm{A}=\left(\mathrm{a}_{1}, \mathrm{a}_{2}\right)$ and $\mathrm{B}=\left(\mathrm{b}_{1}, \mathrm{~b}_{2}\right)$ be two points in the plane with integer coordinates. Which one of the following is not a possible value of the distance between $\mathrm{A}$ and $\mathrm{B}$ ?
- A$\sqrt{65}$
- B$\sqrt{74}$
- Correct answerC$\sqrt{83}$
- D$\sqrt{97}$
Solution
$$
\mathrm{AB}=\sqrt{\left(\mathrm{a}_{1}-\mathrm{b}_{1}\right)^{2}+\left(\mathrm{a}_{2}-\mathrm{b}_{2}\right)^{2}}
$$
Square $+$ Square $=\sqrt{65}$ possible when $=64+1$
$$
\begin{array}{l}
\sqrt{74}=49+25 \\
\sqrt{97}=81+16
\end{array}
$$
But $\sqrt{83}$ not possible
Hence, the correct answer is option: 'C'.