Question
In a plane there are two families of lines $$y = x + r, y = - x + r,$$ where $$r \in \left\{ {0,1,2,3,4} \right\}.$$ The number of squares of diagonals of the length 2 formed by the lines is
A.
9
B.
16
C.
25
D.
None of these
Answer :
9
Solution :
There are two sets of five parallel lines at equal distances. Clearly, lines like $${l_1},{l_3},{m_1},{m_3}$$ form a square whose diagonal’s length is 2.
∴ the number of required squares
$$ = 3 \times 3$$
$$\left\{ {\because \,\,{\text{choices are }}\left( {{l_1},{l_3}} \right),\left( {{l_2},{l_4}} \right),\left( {{l_3},{l_5}} \right)\,{\text{for one set,e}}{\text{.t}}{\text{.c}}{\text{.}}} \right\}.$$
There are two sets of five parallel lines at equal distances. Clearly, lines like $${l_1},{l_3},{m_1},{m_3}$$ form a square whose diagonal’s length is 2.
∴ the number of required squares
$$ = 3 \times 3$$
$$\left\{ {\because \,\,{\text{choices are }}\left( {{l_1},{l_3}} \right),\left( {{l_2},{l_4}} \right),\left( {{l_3},{l_5}} \right)\,{\text{for one set,e}}{\text{.t}}{\text{.c}}{\text{.}}} \right\}.$$