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Question

Find the range of λ if the line 3x4y=λ meets the circle x2+y24x8y5=0 in real points.

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Solution

Eliminating y,
25x22(3λ+80)x+(λ2+32λ80)=0
for real points
B24AC>0 λ2+20λ525<0
or (λ+35)(λ15)<035<λ<15.

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