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Question

If the length of the tangent drawn from (α,β) to the circle x2+y2=6 be twice the length of the tangent from the same point to the circle x2+y2+3x+3y=0, then prove that α2+β2+4α+4β+2=0.

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Solution

Length of tangent from point (α,β) to a circle c=x2+y2+2g+2fy+c=0 equals:α2+β2+2αg+2αf+c
c1=x2+y2=6
c2=x2+y2+3x+3y=0
Tangent length for c1=α2+β26
Tangent length for c2=α2+β2+3α+3β
According to the question,
α2+β26=2α2+β2+3α+3β
α2+β26=4(α2+β2+3α+3β)
0=3α2+3β2+12α+12β+6
0=α2+β2+4α+4β+2

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