Question

求解高中数学坐标系与参数方程

Answer 1

23、

(1) C方程：x=2√3cosα
y=2sinα
x/(2√3)=cosα
y/2=sinα
[x/(2√3)]^2+(y/2)^2=1
C的普通方程：x^2/12+y^2/4=1
x=ρcosθ
y=ρsinθ
(ρcosθ)^2/12+(ρsinθ)^2/4=1
ρ^2cos^2θ+3ρ^2sin^2θ=12
ρ^2cos^2θ+ρ^2sin^2θ+2ρ^2sin^2θ=12
ρ^2+2ρ^2sin^2θ=12
ρ^2(1+2sin^2θ)=12
ρ^2(1+1-cos2θ)=12
C的极坐标方程：ρ^2=12/(2-cos2θ)
(2) A(ρ1,π/6)
B(ρ2,2/3π)
ρ1^2=12/(2-cos2π/6)
=12/(2-1/2)
=8
ρ1=2√2
x=2√2cosπ/6
=2√2(√3/2)
=√6
y=2√2sinπ/6
=2√2(1/2)
=√2
A(√6,√2)
ρ2^2=12/(2-cos4π/3)
=12/(2+1/2)
=24/5
ρ2=2√30/5
x=2√30/5cos(2π/3)
=2√30/5(-1/2)
=-√30/5
y=2√30/5sin(2π/3)
=2√30/5(√3/2)
=3√10/5
B(-√30/5,3√10/5)
|AB|=√[(√6+√30/5)^2+(√2-3√10/5)^2]
=8√5/5