# arrow_back Evaluate $\sqrt{x^{2}+y^{2}+1}=?$

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\begin{aligned}
&x^{2}=17 x+y \\
&y^{2}=x+17 y \\
&x \neq y \\
&\sqrt{x^{2}+y^{2}+1}=?
\end{aligned}

## 1 Answer

Best answer

Best answer
\begin{aligned} &x^{2}=17 x+y(1) \\ &y^{2}=x+17 y(2) \end{aligned}
(1)-(2):
\begin{aligned} &(x-y)(x+y)=16(x-y) \\ &(x-y)(x+y-16)=0 \end{aligned}
since $x \neq y$
then $x+y=16$
(1) +(2):
$x^{2}+y^{2}=18(x+y)=18 \times 16$
Therefor,
$\sqrt{x^{2}+y^{2}+1}=\sqrt{18 \times 16+1}=\sqrt{(17+1)(17-1)+1}=17 .$
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