M(x,y)在圆x^2+y^2=1上移动,求点Q(x(x+y),y(x+y))的轨迹
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M(x,y)在圆x^2+y^2=1上移动,求点Q(x(x+y),y(x+y))的轨迹
用参数方程来表示简单些.
设 M(x,y) 圆的参数方程:
x=cost,y=sint
Q点坐标
x=cost(cost+sint)=cos^2t+cost*sint
=1/2*cos2t+1/2+1/2*sin2t
= √2/2 (√2/2*cost2t + √2/2*sin2t)+1/2
= √2/2 cost(2t-π/4)+1/2
y=sint(cost+sint)=sin^2t+sint*cost
=1/2-1/2*cos2t+1/2*sin2t
=1/2+√2/2 (√2/2*sin2t - √2/2*cos2t)
=√2/2 sin(2t-π/4)+1/2
即Q点的轨迹是以 (1/2,1/2) 为圆心,√2/2 为半径的圆.
设 M(x,y) 圆的参数方程:
x=cost,y=sint
Q点坐标
x=cost(cost+sint)=cos^2t+cost*sint
=1/2*cos2t+1/2+1/2*sin2t
= √2/2 (√2/2*cost2t + √2/2*sin2t)+1/2
= √2/2 cost(2t-π/4)+1/2
y=sint(cost+sint)=sin^2t+sint*cost
=1/2-1/2*cos2t+1/2*sin2t
=1/2+√2/2 (√2/2*sin2t - √2/2*cos2t)
=√2/2 sin(2t-π/4)+1/2
即Q点的轨迹是以 (1/2,1/2) 为圆心,√2/2 为半径的圆.
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