Question

一道关于椭圆的数学题设M是椭圆x平方/25+y平方/16=1上一点,F1 F2为焦点,角F1MF2=30度,求三角形MF

Answer 1

x^2/25+y^2/16=1

焦点F1(-3,0) F2(3,0) |F1F2|=6

由椭圆的定义|F1M|+|F2M|=2a=2*5=10

角F1MF2=π/6

由余弦定理

|F1M|^2+|F2M|^2-2|F1M||F2M|cos(π/6)=|F1F2|^2

|F1M|^2+|F2M|^2-√3|F1M||F2M|=36

(|F1M|+|F2M|)^2-2|F1M||F2M|-√3|F1M||F2M|=36

100-(2+√3)|F1M||F2M|=36

|F1M||F2M|=64/(2+√3)=64(2-√3)

所以S=1/2|F1M||F2M|*sinπ/6=16(2-√3)