过抛物线y2=4x上一点P(4,4),作两条直线分别交抛物线于A(x1,y1),B(x2,y2),直线PA与PB的斜率存在且互为相反数, (1)求y1+y2的值; (2)证明直线AB的斜率是非零常数.
问题描述:
过抛物线y2=4x上一点P(4,4),作两条直线分别交抛物线于A(x1,y1),B(x2,y2),直线PA与PB的斜率存在且互为相反数,
(1)求y1+y2的值;
(2)证明直线AB的斜率是非零常数.
答
(1)设直线PA的斜率为kPA,直线PB的斜率为kPB
由y12=4x1,故kPA=
=
y1−4
x1−4
(x1≠4),4
y1+4
同理可得kPB=
(x2≠4),4
y2+4
由PA,PB斜率互为相反数可得kPA=-kPB,
即
=−4
y1+4
,4
y2+4
化为y1+y2=-8;
(2)设直线AB的斜率为kAB,
由y22=4x2,y12=4x1
∴kAB=
=
y2−y1
x2−x1
=
y2−y1
−y22 4
y12 4
=4
y1+y2
=−4 −8
(常数).1 2