已知p为椭圆x225y2161长轴上一个动点，过点P斜率为k直线交椭圆与两点，若|PA|平方+|PB|平方的值仅仅依赖于k而与P无关，求k值

Question

已知p为椭圆x2 25 y2 16 1长轴上一个动点，过点P斜率为k直线交椭圆与两点，若|PA|平方+|PB|平方的值仅仅依赖于k而与P无关，求k值

Answer 1

P为椭圆x^2/ 25+ y^2/ 16= 1①长轴上一个动点,
∴设P(p,0),-5<=p<=5.
过点P斜率为k直线交椭圆于两点A,B,
设AB:y=k(x-p),②
代入①*400，得16x^2+25k^2(x^2-2px+p^2)=400,
整理得(16+25k^2)x^2-50k^2px+25k^2p^2-400=0,
设A(x1,y1),B(x2,y2),则x1+x2=50k^2p/(16+25k^2),x1x2=(25k^2p^2-400)/(16+25k^2),
|PA|^2+|PB|^2=(x1-p)^2+y1^2+(x2-p)^2+y2^2
=(x1-p)^2+k^2(x1-p)^2+(x2-p)^2+k^2(x2-p)^2(由②)
=[(x1-p)^2+(x2-p)^2](1+k^2)仅仅依赖于k而与P无关,
<==>(x1-p)^2+(x2-p)^2=x1^2+x2^2-2p(x1+x2)+2p^2
=(x1+x2)^2-2x1x2-2p(x1+x2)+2p^2
=(x1+x2)(x1+x2-2p)-2x1x2+2p^2
=[50k^2p(50k^2p-32p-50k^2p)-2(25k^2p^2-400)(16+25k^2)+2p^2(256+800k^2+625k^4)]/(16+25k^2)^2
=[-1600k^2p^2-2(400k^2p^2+625k^4p^2-6400-10000k^2)+2(256p^2+800k^2p^2+625k^4p^2)]/(16+25k^2)^2
=-2（400k^2p^2-6400-10000k^2-256p^2)/(16+25k^2)^2与p无关，
∴上式中p^2的系数400k^2-256=0,k^2=16/25,k=土4/5.