y=x+r and y= -x+r where r takes all decimal digits. Then the number of squares in xyplane formed by these lines with diagonals of 2 units length are a

(A)81 (B) 100
(C) 64 (D)49

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{Let} \mathrm{side} \mathrm{of} \mathrm{diagonal} \mathrm{be} \mathrm{a} \mathrm{units}. \\ \Rightarrow \mathrm{Diagonal}=\mathrm{a}\sqrt{2} \mathrm{units} \\ \Rightarrow \mathrm{a}\sqrt{2}=2 \left[\mathrm{Given}\right] \\ \Rightarrow \mathrm{a}=\sqrt{2} \mathrm{units} \\ \mathrm{y}=\mathrm{x}+\mathrm{r} ;\left(\mathrm{i}\right) \\ \mathrm{y}=-\mathrm{x}+\mathrm{r} \left(\mathrm{ii}\right) \\ \left(\mathrm{i}\right)+\left(\mathrm{ii}\right), \mathrm{we} \mathrm{get} \\ 2\mathrm{y}=2\mathrm{r} \\ \Rightarrow \mathrm{y}=\mathrm{r} \\ \mathrm{Put} \mathrm{it} \mathrm{in} \left(\mathrm{i}\right) \\ \mathrm{r}=\mathrm{x}+\mathrm{r} \\ \Rightarrow \mathrm{x}=0 \\ \mathrm{Hence} \mathrm{point} \mathrm{of} \mathrm{intersection} \mathrm{is} \left(0,\mathrm{r}\right) \mathrm{i}.\mathrm{e} \mathrm{on} \mathrm{Y}-\mathrm{axis}. \end{gathered}$$

$$\begin{gathered} \mathrm{Now} \mathrm{you} \mathrm{can} \mathrm{clearly} \mathrm{see} \mathrm{that} \mathrm{diagonal} \mathrm{will} \mathrm{either} \mathrm{be} \mathrm{vertical} \mathrm{or} \\ \mathrm{horizontal}. \\ \mathrm{Given} \mathrm{that} \\ \mathrm{we} \mathrm{have} \mathrm{r}\in \left\{0,1,2,3,4,5,7,8,9\right\} \end{gathered}$$

$$\begin{gathered} \mathrm{Start} \mathrm{with} \mathrm{the} \mathrm{point} \mathrm{A}=\left(-4.5,4.5\right) \mathrm{and} \mathrm{consider} \mathrm{the} \mathrm{point} \mathrm{B}=\left(-2.5,4.5\right). \\ \mathrm{There} \mathrm{you} \mathrm{have} \mathrm{a} \mathrm{diagonal}. \mathrm{Now} \mathrm{make} \mathrm{that} \mathrm{diagonal} \mathrm{glide} \mathrm{to} \mathrm{reach} \mathrm{the} \\ \mathrm{points} \mathrm{C} \mathrm{and} \mathrm{D}. \mathrm{Do} \mathrm{this} \mathrm{until} \mathrm{you} \mathrm{reach} \mathrm{the} \mathrm{points} \mathrm{E} \mathrm{and} \mathrm{F}. \\ \mathrm{You} \mathrm{have} \mathrm{exactly} 8 \mathrm{diagonals}. \mathrm{Start} \mathrm{all} \mathrm{over} \mathrm{again} \mathrm{with} \mathrm{point} \mathrm{G} \mathrm{and} \mathrm{H} \\ \mathrm{and} \mathrm{make} \mathrm{them} \mathrm{glide}: \mathrm{again} 8 \mathrm{diagonals}. \mathrm{You} \mathrm{go} \mathrm{on} \mathrm{like} \mathrm{this} \mathrm{until} \mathrm{you} \\ \mathrm{reach} \mathrm{I} \mathrm{and} \mathrm{J}, \mathrm{where} \mathrm{you}\text{'}\mathrm{ll} \mathrm{be} \mathrm{gliding} \mathrm{the} \mathrm{diagonal} \mathrm{for} \mathrm{the} \mathrm{last} \mathrm{time}. \\ \mathrm{Hence} \mathrm{there} \mathrm{are} \mathrm{exactly} 8\times 8=64 \mathrm{such} \mathrm{squares}. \\ \mathrm{Hence} \mathrm{option}\left(\mathrm{C}\right) \mathrm{is} \mathrm{correct}. \end{gathered}$$

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