the locus of a point P which divides line joining (1, 0) and (2 cos theta, 2 sin theta )internally in the ratio 1:2 is a
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Let the coordinate of P is (x,y). Then by the given question,
x=(1+2cosθ)/(1+2)=(1+2cosθ)/3
or, 3x=1+2cosθ
or, 2cosθ=3x-1
or, cosθ=(3x-1)/2 and
y=(0+2sinθ)/(1+2)=2sinθ/3
or, 3y=2sinθ
or, sinθ=3y/2
We know that, sin²θ+cos²θ=1
or, (3y/2)²+(3x-1)²/2²=1
or, 9y²/4+(3x-1)²/4=1
or, (3x-1)²+9y²=4 which is a circle.
x=(1+2cosθ)/(1+2)=(1+2cosθ)/3
or, 3x=1+2cosθ
or, 2cosθ=3x-1
or, cosθ=(3x-1)/2 and
y=(0+2sinθ)/(1+2)=2sinθ/3
or, 3y=2sinθ
or, sinθ=3y/2
We know that, sin²θ+cos²θ=1
or, (3y/2)²+(3x-1)²/2²=1
or, 9y²/4+(3x-1)²/4=1
or, (3x-1)²+9y²=4 which is a circle.
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