Question

设F1F2分别是椭圆x²/25+y²/16=1的左右焦点，P为椭圆上一点M是F1P的中点|OM|=3

设F1F2分别是椭圆x²/25+y²/16=1的左右焦点，P为椭圆上一点M是F1P的中点|OM|=3则P点到椭圆左焦点的距离为

Answer 1

a=5；b=4；c=3；∣OM∣=3，∣OF₁∣=3；F₁(-3，0);
设P点的坐标为(5cost， 4sint)；M点的坐标为(x，y);
M是F₁P的中点；故x=(5cost-3)/2；y=(4sint)/2;
∣OF₁∣=√(x²+y²)=√[(5cost-3)²/4+(4sint)²/4]=3
∴25cos²t-30cost+9+16sin²t=36
9cos²t-30cost+25=36；即9cos²t-30cost-11
=(3cost+1)(3cost-11)=0
∴cost₁=-1/3；cost₂=-11/3(舍去).
sint₁=√(1-1/9)=(2/3)√2;
∴x=(5cost-3)/2=(-5/3-3)/2=-7/3; y=(4sint)/2=(4/3)√2；
∴∣F₁P∣=2∣F₁M∣=2√[(x+3)²+y²]=2√[(-7/3+3)²+2(4/3)²]
=2√[(4/9)+(32/9)]=2√(36/9)=2×(6/3)=4.