9在平面直角坐标系中,已知点P在第三象限角的平分线上,且到x轴的距离为2,则当ρ>0,θ∈[0,2π)时,点P的极坐标为　　　　　.

解析∵点P(x,y)在第三象限角的平分线上,且到x轴的距离为2,

∴x=-2,且y=-2.∴ρ=√(x^2+y^2 )=2√2.

又tanθ=y/x=1,且θ∈[0,2π),∴θ=5π/4.

∴点P的极坐标为(2√2 "," 5π/4).

答案(2√2 "," 5π/4)

10(1)已知点的极坐标分别为A(3",-" π/4),B(2"," 2π/3),C(√3/2 "," π),D(4"," 3π/2),求它们的直角坐标;

(2)已知点的直角坐标分别为A(√6,√2),B(0",-" √5/3),C(-2,-2√3),求它们的极坐标(ρ>0,0≤θ<2π).

解(1)根据x=ρcosθ,y=ρsinθ,得A((3√2)/2 ",-" (3√2)/2),B(-1,√3),C("-" √3/2 "," 0),D(0,-4).

(2)根据ρ2=x2+y2,tanθ=y/x(x≠0),结合各点所在的象限得A(2√2 "," π/6),B(√5/3 "," 3π/2),C(4"," 4π/3).

能力提升

1若ρ1=ρ2≠0,θ1-θ2=π,则点M1(ρ1,θ1)与点M2(ρ2,θ2)的位置关系是(　　)

A.关于极轴所在的直线对称

B.关于极点对称

C.关于过极点且垂直于极轴的直线对称

D.重合

答案B

★2在极坐标系中,已知点P(2"," π/3)和点Q(2√3 "," 5π/6),则线段PQ的中点M的极坐标是(　　)

A.(2"," π/3)B.(2"," 2π/3)

C.(1+√3 "," 7π/12)D.(1+√3 "," 5π/12)

解析∵P(2"," π/3),∴{■(x=2cos π/3=1"," @y=2sin π/3=√3 "," )┤

∴P(1,√3).

∵Q(2√3 "," 5π/6),∴{■(x=2√3 cos 5π/6="-" 3"," @y=2√3 sin 5π/6=√3 "," )┤