Question

The graph of the equation ${x^2} + {y^2} = 25$ includes how many points $\left( {x,y} \right)$ in the coordinate plane where $x$ and $y$ are both the integers.

Answer 1

Hint: We will first consider the equation of the circle; we need to find how many points can be included in the graph of the circle. So, we will use a hit and trial method to choose the values of \[x\] and \[y\] and in this way we will find the number of points which will satisfy the equation and thus be included in the graph.

Complete step by step solution:
First consider the given equation of circle that is ${x^2} + {y^2} = 25$
We have to determine the number of points which is included in the graph,
Thus, we will try different integers which will satisfy the equation ${x^2} + {y^2} = 25$.
We will first consider the point \[\left( {3,4} \right)\] where $x = 3$ and $y = 4$.
Thus, we will substitute these values in equation ${x^2} + {y^2} = 25$,
$
   \Rightarrow {3^2} + {4^2} = 25 \\
   \Rightarrow 9 + 16 = 25 \\
   \Rightarrow 25 = 25 \\
 $
Hence, we get the one solution of the given equation,
Now, we will find the other solutions of this equation, which are $\left( {4,3} \right),\left( { - 3, - 4} \right){\text{and }}\left( { - 4, - 3} \right)$ as all three points will satisfy the equation ${x^2} + {y^2} = 25$.
Thus, we get the 4 points which satisfies the equation ${x^2} + {y^2} = 25$.
So, the option A is correct.

Note: We know the general form of the equation of circle that is ${x^2} + {y^2} = {r^2}$. To find the points which will satisfy the equation, we have to check those points which on squaring gives us 25. And the negative coordinates also become positive on squaring so, in this way we get 4 coordinates which will be included in the graph. To verify the answers we can also make the graph and check whether the points are lying in the circle or not.