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

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 .