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Question

The number of points $$ \displaystyle \left ( x,y \right ) $$ having integral co-ordinates satisfying the condition $$ \displaystyle x^{2}+y^{2}< 25 $$ is


A
81
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B
12
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C
66
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D
69
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Solution

The correct option is D $$69$$
Since $$ { x }^{ 2 }+{ y }^{ 2 }<25 $$ and $$ a$$ and $$y$$ are integers, the 
possible values of $$ x$$ and $$ y\in \left( 0,\pm 1,\pm 2,\pm 3,\pm 4, \right)  $$. 
Thus,$$x$$ and $$y$$ can be chosen in $$9$$ ways each and $$(x,y) $$ can be chosen in $$9\times 9=81 $$ ways.
However, we have to exclude cases $$\left( \pm 3,\pm 4 \right) ,\left( \pm 4,\pm 3 \right)  $$
and $$ \left( \pm 4,\pm 4 \right) i.e.,3\times 4=12 $$
Hence, the number of permissible values$$=81-12=69. $$

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