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arrow_back Evaluate \(\sqrt{x^{2}+y^{2}+1}=?\)

by Gold Status
(30,543 points)
in Mathematics
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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}

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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,
\[
\sqrt{x^{2}+y^{2}+1}=\sqrt{18 \times 16+1}=\sqrt{(17+1)(17-1)+1}=17 .
\]
by Gold Status
(30,543 points)

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