Question

Find the number of points \[\left( {x , y} \right)\]having integral coordinates satisfying the condition${x^2} + {y^2} < 25$.
\[1) 81\]
\[2) 12\]
\[3) 66\]
\[4) 69\]

Answer 1

Hint: We need to find the number of points which satisfy the condition${x^2} + {y^2} < 25$. We solve this question by using the integer values of $x$ and $y$ such that all the values lie inside the figure of the given equation . We should simply know how to relate the values of the points in the given equation .

Complete step-by-step solution:
Given :
${x^2} + {y^2} < 25$
$x$ and $y$ are integers
As, we know that $x$ and $y$ are integers
For all the possible values we will put different values of and in the given condition . The integer values satisfying the given conditions would be the possible values for the solution .
let \[x = 0\] and putting different of \[y\] we will check the condition :
\[y = \pm 1\]
\[{0^2} + {\left( { \pm 1} \right)^2} < 25\]
On solving , we get
\[0 + 1 < 25\]
\[1 < 25\]
True , so y=+_1 are solutions
\[y = \pm 2\]
\[{0^2} + {\left( { \pm 2} \right)^2} < 25\]
On solving , we get
\[0 + 4 < 25\]
\[4 < 25\]
True , so \[y = \pm 2\] are solutions
\[y = \pm 3\]
\[{0^2} + {\left( { \pm 3} \right)^2} < 25\]
On solving , we get
\[0 + 9 < 25\]
\[9 < 25\]
True , so \[y = \pm 3\] are solutions
\[y = \pm 4\]
\[{0^2} + {\left( { \pm 4} \right)^2} < 25\]
On solving , we get
\[0 + 16 < 25\]
\[16 < 25\]
True , so \[y = \pm 4\] are solutions
\[y = \pm 5\]
\[{0^2} + {\left( { \pm 5} \right)^2} < 25\]
On solving , we get
\[0 + 25 < 25\]
\[25 < 25\]
False , so \[y = \pm 5\] are not solutions
As squaring a positive number or a negative number gives the same result that’s why both are shown together .
These are the possible values of \[x\] and \[y\] . For all the integers greater than \[5\] , the values won’t satisfy the given condition .
[Need not to solve for \[x\] , as it will also yield the same result as that of \[y\]]
Hence , All the possible values of $x$ and $y$ are\[\left\{ { 0 , \pm 1 , \pm 2 , \pm 3{\text{ ,}} \pm 4 } \right\}\]
So ,
The values of $x$ and $y$ can be chosen in $9$ ways each
Then , we get the number of ways as
\[\text{The total number of ways} = \text{Number of values of} \times \text{Number of values of y}\]
the total number of ways \[ = 9 \times 9\]
the total number of ways \[ = 81 ways\]
Also , if both $x$ and $y$ are equal to\[ \pm 4\], then the condition will not be satisfied .
So , the number of ways for which the equation is not satisfied are :- \[\left( { + 3 , \pm 4 } \right) or \left( { + 4 , \pm 3 } \right) or \left( { \pm 4 , \pm 4 } \right)\]
The total ways in this the equation is not satisfied \[ = 3 \times 4\]
The total ways in this the equation is not satisfied \[ = 12 ways\]
On solving ,
The total number of integral points of \[\left( { x , y } \right) = 81 - 12\]
The total number of integral points of\[\;\left( { x , y } \right) = 69 ways\]
Thus , the total integral points having integral coordinates are $69$ .
Hence , the correct option is\[\left( 4 \right)\].

Note: The given equation is the equation of a circle with radius 5 units and with centre points \[\left( { 0 , 0 } \right) .\]Finding the integral points which satisfy the condition ${x^2} + {y^2} < 25$ means all the points of \[\left( { x , y } \right)\] which lie inside the circle.
As the points are integral that is why we have chosen only these $9$ values . Had it been a natural number then the possible points would have been : - \[\left\{ { 1 , 2, 3, 4 } \right\}\]only .