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www.999818.com:江苏省2015届高考数学解答题每日一练系列——应用题,解析几何3

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江苏高考数学解答题每日一练——应用题, 解析几何

编号: 003

1.铁路线上的 AB 段长 100 公里,工厂 C 到铁路的距离 CA 为 20 公理,已知铁路每吨公里与公路每吨公里的运费 之比为 3∶ 5,为了使原料从供应站 B 运到工厂 C 的运费最省,D 点应改在何处?

2 y2 0) ,离心率为 2 . 2、如图,在平面直角坐标系 xOy 中,椭圆 x 2 ? 2 ? 1(a ? b ? 0) 的右焦点为 F (1, 2 a b

分别过 O , F 的两条弦 AB , CD 相交于点 E (异于 A , C 两点) ,且 OE ? EF . (1)求椭圆的方程; (2)求证:直线 AC , BD 的斜率之和为定值.

y
C

A E
O

F

D

x

B
(第 18 题)

江苏高考数学解答题每日一练——应用题, 解析几何

编号: 003
2

1、解法一:设 AD=x, 则 BD=100-x, CD= 400+x ,设总运费为 y, 则 y=3k(100-x)+5k 400+x2=k(300-3x+5 400+x2)=k[300+(4 400+x2-4x)+( 400+x2+x)] =k[300+ 4×400 400+x +x 4×400 400+x +x
2 2

+( 400+x2+x)]≥k[300+2

·( 400+x2+x)]=380k。 400+x2+x

4×400

当且仅当

= 400+x2+x 时取等号。解此方程,求得=15 公里。

即当点 D 距 A 为 15 公里处时运费最低。 y 解法二:同解法二先得 y=3k(100-x)+5k 400+x2设 -300=t, 则 t=-3x+5 400+x2>0 k 则 (t+3x)2=25(400+x2) t2+6xt+9x2=25× 400+25x2 16x2-6tx+400× 25-t2=0

Δ =36t2-4× 16× (400× 25-t2)≥0 即 t2≥16×400 ∵ t>0 ∴ t≥4×20=80, ∴ 当且仅当,t=80 时,y 最小。 b 6×80 此时,x=- = =15 ∈ (0,100)。 2a 2×16 即当点 D 距 A 为 15 公里时,运费最低。 2、 (1)解:由题意,得 c ? 1 , e ? c ? 2 ,故 a ? 2 , a 2 从而 b2 ? a 2 ? c 2 ? 1 , 所以椭圆的方程为 x ? y 2 ? 1 . 2 (2)证明:设直线 AB 的方程为 y ? kx , ② ………7 分
2



………5 分

直线 CD 的方程为 y ? ?k ( x ? 1) , ③ 由①②得,点 A , B 的横坐标为 ?

2 , 2k 2 ? 1
………9 分

由①③得,点 C , D 的横坐标为

2k 2 ? 2(k 2 ? 1) , 2k 2 ? 1

kx1 ) , B( x2, kx2 ) , C ( x3, k (1 ? x3 )) , D( x4, k (1 ? x4 )) , 记 A( x1,
则直线 AC , BD 的斜率之和为

kx1 ? k (1 ? x3 ) kx2 ? k (1 ? x4 ) ( x ? x ? 1)( x2 ? x4 ) ? ( x1 ? x3 )( x2 ? x4 ? 1) ? ?k? 1 3 x1 ? x3 x2 ? x4 ( x1 ? x3 )( x2 ? x4 )
2 ?2 ? 2(k ? 1) ? ? 0 ? 4k 2 2? ? ? 2 2 2( x x ? x x ) ? ( x1 ? x2 ) ? ( x3 ? x4 ) 2k 2 ? 1 ? 0 ? k ? ? 2 k ? 1 2k ? 1 ? ?k? 1 2 3 4 ( x1 ? x3 )( x2 ? x4 ) ( x1 ? x3 )( x2 ? x4 )




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