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问题 解答题
已知O为坐标原点，
OA
=(-4，0)，
AB
=(8，0)，动点P满足|
PA
|+|
PB
|=10
（1）求动点P的轨迹方程；
（2）求
PA
•
PB
的最小值；
（3）若Q（1，0），试问动点P的轨迹上是否存在M、N两点，满足
NQ
=
4
3
QM
？若存在求出M、N的坐标，若不存在说明理由．
答案

解（1）∵

OA
=(-4，0)，
AB
=(8，0)，

∴A（-4，0），B（4，0）．

又∵动点P满足|

PA
|+|
PB
|=10，

∴动点P的轨迹为以A，B为焦点的椭圆，且长轴长2a=10∴a=5，b=3．

椭圆方程为

x2
25
+
y2
9
=1．

（2）

PA
•
PB
=|
PA
||
PB
|cos∠APB=|
PA
||
PB
|
|
PA
|2+|
PB
|2-4c2
2|
PA
||
PB
|
=2a2-2b2-|
PA
||
PB
|=18-|
PA
||
PB
|≥18-
 (|
PA
|+|
PB
|)2
4
=-7，∴
PA
•
PB
的最小值为-7

（3）假设存在M、N两点，满足

NQ
=
4
3
QM
，则M，Q，N共线，

设M（x1，y1），N（x2，y2），由

NQ
=
4
3
QM
，可得
1-x2=
4
3
x1-1
-y2=
4
3
y1
，∴y2=-
4
3
y1．①

设方程为x=my+1，代入椭圆方程，化简得，（9m2+25）y2+18my-216=0，

y1+y2=-

18m
9m2+25
，y1y2=-
216
9m2+25
，把①代入，得y1=
54m
9m2+25
，y12=
162
9m2+25

∴m=

5
3
或-
5
3

单项选择题

铺设预制混凝土块路面，在修理平整、调整缝宽后，应用下列哪种材料灌缝？（）

A.素水泥浆灌缝

B.先灌干砂，再洒水使其沉实

C.水泥砂浆灌缝

D.填塞木丝板条
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单项选择题

It is possibly the most famous promise in British retailing: "Never knowingly undersold" has been at the heart of John Lewis’s business since 1925. But a quietly introduced change has infuriated loyal customers, who claim the price-match promise is now slipping away. For many years John Lewis customers have been safe in the knowledge that if they found their purchase for a lower price elsewhere the company would refund the difference. Carrier bags and marketing campaigns have proudly proclaimed to the world that John Lewis won’t be beaten on price.
Yet since September some customers who have asked John Lewis to match the price of goods found cheaper elsewhere on the high street have been turned away. A Guardian Money reader from Roydon, Essex, contacted us after he bought a Hotpoint washing machine in John Lewis’s Welwyn store for ￡279. A few days later he saw the same model in Argos for ￡219--￡60 cheaper. John Lewis turned down his claim made under the never knowingly undersold policy, because it said it guaranteed the washing machines for two years, while Argos offered only one year.
The customer complained--unsuccessfully--that the store wasn’t being fair as this was not made clear in the literature. When Money investigated, we found that John Lewis had made a fundamental change to its policy. In a statement in September, which at the time drew positive headlines, it said it would for the first time match online prices from other retailers as long as they also had a physical high street presence. What was made less clear was that the store would no longer match a price unless its rival offers the exact same warranty.
The policy change might not sound much, but it in effect allows the store to avoid almost all price matching of electrical items--because John Lewis has adopted a policy of offering two-year warranties on almost every such item. Most stores in the UK offer just one year. When we first raised the reader’s complaint with John Lewis it told us. "As part of our commitment to be never knowingly undersold, we match prices based on the combined cost of the product plus charges the competitor may make for a comparable warranty or guarantee. We evaluate price-match claims on a like-for-like basis, and breakdown cover is a crucial part of our proposition to our customers. "
What it failed to mention was that prior to the September policy rewrite, it would have paid the complainant the ￡60 difference between the John Lewis and Argos washing machines. Interestingly, the store confirmed it would not price match the cost of buying a product plus a warranty from a third party company, but would consider a claim if the cheaper retailer offered the chance to buy both together.
David Suddock, head of buying support at John Lewis, who revised the policy, says. "As a result of our commitment to expand our never knowingly undersold policy to include other retailers with online presences we now put a great deal of resources into checking the prices charged by our rivals and lowering ours where appropriate; Our customers are benefiting through significantly reduced prices. They tell us they value the extra warranty periods we offer, and we think it is only fair we should include that in our price match scheme. The terms of the never knowingly undersold policy are clearly presented in both our stores and on the website. "
But if the Money postbag is to be believed, most John Lewis customers were unaware of the change. And Martyn Hocking, editor of Which says: "John Lewis is known for its great customer service, so the change to its never knowingly undersold policy is very disappointing. Customers would naturally expect any price matching policy to relate to the up-front cost of a product, excluding the value of added extras such as warranties and guarantees. As such, we feel that the amended policy is misleading and will lead to frustration for many shoppers. "
But Natalie Berg, research director at retail analysts Planet Retail, says John Lewis\s move was perhaps inevitable. "The internet has put the power to compare prices in the hands of all of us; some shoppers now use smart phones to check prices as they walk around a store. John Lewis has realized that while price is important, it’s not the factor in where to buy. The fact that John Lewis has been one of the winners on the high street in recent months suggests consumers are not just looking for the lowest prices, but they want value--and the perception that John Lewis delivers this. \

The ￡60 difference between the John Lewis and Argos washing machines (para. 2)______.

A. has been paid to the customer after the complaint
B. would not be paid according to the September policy rewrite
C. should be paid to the customer according to the original price match policy
D. is already covered in the two year guarantee
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