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C

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$P=1-\dfrac{1}{y^2}-\dfrac{1}{x^2}+\dfrac{1}{x^2y^2}$

\Leftrightarrow $P=1+\dfrac{1-(x^2+y^2)}{x^2y^2}$

\Leftrightarrow $P=1+\dfrac{1-(x+y)^2+2xy}{x^2y^2}=1+\dfrac{2}{xy}$

\geq $1+\dfrac{2}{\dfrac{1}{4}}=9$

(do theo Cauchy $xy$ \leq $\dfrac{(x+y)^2}{4}=\dfrac{1}{4}$)

$P_{min}=9$ \Leftrightarrow $x=y=\dfrac{1}{2}$
 
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