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Question

If variable parameter α,β,γR and satisfy the relations α2+2β2+3α=1+γ2 and 2α2+4β2=2γ2+5β, then

A
Minimum value of α2+β2 is 461
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B
Minimum value of α2+β2 is 217
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C
Minimum value of α2+β22α+1 is 1661
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D
Minimum value of α2+β22α+1 is 117
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Solution

The correct options are
A Minimum value of α2+β2 is 461
C Minimum value of α2+β22α+1 is 1661
α2+2β2+3α=1+γ2 (1)
2α2+4β2=2γ2+5β (2)
From equation (1)×2 equation (2), we get
6α+5β=2

Perpendicular distance from centre (0,0) to line 6x+5y=2 is |2|36+25
Hence, the minimum value of α2+β2 is (|2|36+25)2=461

Perpendicular distance from centre (1,0) to line 6x+5y=2 is |62|36+25
Hence, the minimum value of α2+β22α+1 is (|62|36+25)2=1661

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