\ begin {frame} {2019 in Beijing in the college entrance examination}
\ the begin {exampleblock} {2019 in Beijing in Science College Entrance Examination}
mathematics, there are many beautiful shape, meaning beautiful curves, curve $ C: x ^ 2 + y ^ 2 = 1 + | x | y $ is one (FIG) gives the following three conclusions:.
\ the enumerate the begin {%} [align = left = left, labelsep = -0.6em, leftmargin = 1.2em, noitemsep, topsep = 0pt]
\ item [\ ding {172}] curve just after $ C $ $ $ integer-points 6 (i.e., the horizontal and vertical coordinates of a point are integers);
\ Item [\ ding {173}] $ C $ arbitrary curve point to the origin of the distance does not exceed $ \ sqrt {2} $;
Area \ item [\ ding {174} ] curve $ C $ surrounded by the "heart-shaped" region is less than $. 3 $.
\ End {the enumerate}
wherein all the correct conclusion number is
\ the begin {Tasks} (2)
\ Task \ Ding {172}
\ Task \ Ding {173}
\ Task \ Ding {172} \ Ding {173}
\ Task \ Ding {172} \ Ding {173} \ Ding {174}
\ End {Tasks}
\ End { exampleblock}
\ textbf {Solution.} In fact, the use of mean apparent inequality
\ [1+ | x | y = x ^ 2 + y ^ 2 \ geq 2 | xy |. \]
If $ y \ geq 0 $ is an integer, $ 1 + | x | y \ geq 2 | x | y $, then $ | x | y \ leq 1 $ $ x = 0 $ when time, $ (x, y) = (0,1) $;. when the $ x \ when neq 0 $, $ y = 0 $ must $ 1 or $, then $ (x, y) = ( - 1,0), (1,0), (- 1,1) $ or $ (1, 1) $.
If $ y <0 $ is an integer, $ 1 + | x | y \ geq -2 | x | y $, then the $ -3 | x | y \ leq 1 $, $ 0 certainly when $ x =, then $ (x, y) = (0, -1) $.
In summary, the curve just after $ C $ 6 $ $ integer-point $ (x, y) = ( 0, -1), (0,1), (- 1,0), (1,0), ( -1,1) $ or $ (1,1) $, is the figure $ a, B, C, D , E, F $ six points. therefore, the option \ ding {172} right.
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\ begin {frame} {123}
polar coordinates substitution $ x = r \ cos \ theta , y = r \ sin \ theta $ $ C $ curve may equations $ x ^ 2 + y ^ 2 = 1 + | x | y $ rewritten as $ r ^ 2 = 1 + r ^ 2 | \ cos \ theta | \ sin \ theta $, so
\ [r ^ 2 = \ frac {1} {1- | \ cos \ theta | \ sin \ theta} \ leq \ frac {1} {1- | \ cos \ theta \ sin \ theta |} = \ frac {1} {1- \ frac {1} {2} | \ sin (2 \ theta) |} \ leq 2, \]
Thus $ r \ leq \ sqrt {2 } $, i.e., any distance to the origin point does not exceed $ \ sqrt {2} $ C $ $ curve, as shown, to the origin O $ $ radius may cover a circle of the heart-shaped, $ $ is the OB, so curve $ C $ $ B (1,1) $ at a maximum distance from the origin to the $ \ sqrt {2} $. Therefore option \ Ding {173} correctly.
And because the area of the heart-shaped region is larger than the polygon ABCDEF $ $ area, i.e. greater than $ 3 $. Therefore option \ ding {174} error.
Therefore, the correct answer is (C).
Further, we can use calculus University Mathematics the exact value of an area of the heart-shaped:
\ [S = 2 \ int _ {- \ FRAC {\ PI} {2}} ^ {\ FRAC {\ PI} {2 }} {\ frac {1} {2} \ left (\ frac {1} {1- \ cos \ theta \ sin \ theta} \ right) ^ 2d \ theta} = \ frac {8 \ pi} {3 \ {}}. 3 sqrt \ approx 4.8368 \ cdots \]
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