If variable parameter α,β,γ∈R 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+β2−2α+1 is 1661
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D
Minimum value of α2+β2−2α+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+β2−2α+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 |6−2|√36+25 Hence, the minimum value of α2+β2−2α+1 is (|6−2|√36+25)2=1661