Question

Determine all ordered pairs that satisfy $(x-y{)}^2+x^2=25$, where $x$ and $y$ are integers and $x\ge 0$. Find the number of different values of $y$ that occur

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

The correct option is B $4$

${(x-y)}^2+x^2=25$
Now, $3^2+4^2=5^2$
$9+16=25$
$\therefore$There are 2 possibilities:
$I.{(x-y)}^2=9$ and $x^2=16$
$\therefore x=\pm 4$ and $x-y=\pm 3$
$\left(i\right).x-y=3\Rightarrow (4,1)$ and $(-4,-7)$
$\left(ii\right).x-y=-3\Rightarrow (4,7)$ and $(-4,-1)$
$II.{(x-y)}^2=16$ and $x^2=9$
$\therefore x=\pm 3$ and $x-y=\pm 4$
$\left(i\right).x-y=4\Rightarrow (3,-1)$ and $(-3,-7)$
$\left(ii\right).x-y=-4\Rightarrow (3,7)$ and $(-3,-1)$
$\therefore$ Different values of y are $1,-1,7,-7$
$\therefore 4$ different values of y occur.