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问题 填空题
已知A是曲线C1：y=
a
x-2
（a＞0）与曲线C2：x2+y2=5的一个公共点．若C1在A处的切线与C2在A处的切线互相垂直，则实数a的值是______．
答案

设点A的坐标为（x0，y0），代入两曲线方程得：

y0=

a
x0-2
①，x02+y02=5②，

由曲线C1：y=

a
x-2
得：y′=-
a
(x-2)2
，

则曲线C1在A处的切线的斜率k=-

a
(x0-2)2
，

所以C1在A处的切线方程为：y=-

a
(x0-2)2
（x-x0）+y0，

由C1在A处的切线与C2在A处的切线互相垂直，

得到切线方程y=-

a
(x0-2)2
（x-x0）+y0过圆C2的圆心（0，0），

则有-

a
(x0-2)2
（0-x0）+y0=0，即y0=-
ax0
(x0-2)2
③，

把③代入①得：

a
x0-2
=-
a
(x0-2)2
x0从而x0=1再代入①得：y0=-a；代入②，

得：1+a2=5（a＞0）．

则a=2（-2舍去）．

故实数a的值为2．

填空题

[A] Storytellers from antiquity knew the power of star-crossed romance, and so did Audrey Niffenegger. Her 2003 best seller The Time Traveler’s Wife is so plangent a tale of fatal love, with two adorable people fighting to beat the odds against them.
[B] Henry, you see, has the gift or curse of time-traveling: disappearing from one temporal and spatial reality to pop up, naked, in another. This science-fiction trope will be familiar to fans of The Terminator, but Henry is no action-fantasy god. He’s just a guy whose body has a wanderlust he can’t harness. That’s why, as he tells the besotted Clare, "I never wanted anything in my life that I couldn’t stand losing. " Of course they’re destined to be each other’s one and only loves.
[C] My friend and neighbor, the filmmaker Alan Wade, has a provocative explanation for why Titanic struck such a p and reverberant chord with hundreds of millions of moviegoers, especially women: the hero dies. OK, that breaks a cardinal rule of movie romance: that the lovers kiss happily at the final fade-out. Most examples of the genre end with that rosy image, in part because their makers are reluctant to bum out their audience.
[D] Henry (Eric Bana), who works in a Chicago public library, is in the reading room when a woman he’s never met walks up to him and says dewily, "I’ve loved you all my life. " She’s Clare (Rachel McAdams), a young artist, and in her past--Henry’s future-he has visited her and won her undying devotion.
[E] James Cameron must have been tempted to end his film with Leonardo DiCaprio’s Jack surviving the ship’s sinking and enjoying a long life with Kate Winslet’s Rose.
[F] That it’s surprising it took six years for it to get to the big screen. Maybe prospective producers were reluctant to buck the prevailing wisdom of a conventional happy ending. Anyway, here is the film version, directed by Robert Schwentke. It’s soppy enough to suit the requirements of the weepie genre, and there’s a music score that tries to cue all the emotions in viewers, as if they’re incapable of locating their own feelings. But the movie also has an aching solidity that allows you to surrender to its cuddly-creepy feelings without hating yourself in the morning.
[G] But Cameron realized that by killing off Jack, he was raising the movie’s stakes from domestic platitude to classic romantic tragedy. Jack’s death stamped both finality and immortality on the lovers’ shipboard tryst. Because he is gone, their love will live forever.
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