已知双曲线的标准方程为x^2/9-y^2/16=1,F为其右焦点,A1,A2是实轴的两端点,设P为双曲线上不同于A1,A2的任意一点,直线A1P,A2P与直线x=a分别交于M,N若FM向量与FN向量的数量积=0,则a的值为.

Question

已知双曲线的标准方程为x^2/9-y^2/16=1,F为其右焦点,A1,A2是实轴的两端点,设P为双曲线上不同于A1,A2的任意一点,直线A1P,A2P与直线x=a分别交于M,N若FM向量与FN向量的数量积=0,则a的值为.

Answer 1

双曲线 x^2/9-y^2/16=1,
右焦点F(5.0),A1(-3,0),A2(3,0)
设P(x,y) M (a,m),N(a,n)
∵P,A1,M三点共线,
∴m/(a+3)=y/(x+3)
∴m=y(a+3)/(x+3)
∵P,A2,N三点共线,
∴n/(a-3)=y/(x-3)
∴n=y(a-3)/(x-3)
∵x^2/9-y^2/16=1
∴(x^2-9)/9=y^2/16
∴y^2/(x^2-9)=16/9
FM向量=(a-5,y(a+3)/(x+3))
FN向量=(a-5,y(a-3)/(x-3))
FM向量*FN向量
=(a-5)^2+y^2(a^2-9)/(x^2-9)
=(a-5)^2+16(a^2-9)/9
∵FM向量*FN向量=0
∴(a-5)^2+16(a^2-9)/9=0
25a^2-90a+81=0
∴a=9/5