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Question

Let (a,b) be a point on a circle which passes through (3,1) and touches the line x+y=2 at the point (1,1). If maximum possible value of a is α, then a quadratic equation with rational coefficients whose one root is α, is

A
x2+2x9=0
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B
x2+2x7=0
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C
x2+2x6=0
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D
x2+2x8=0
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Solution

The correct option is B x2+2x7=0
Equation of the circle touching the line x+y2=0 at the point (1,1) is given by
S:(x1)2+(y1)2+λ(x+y2)=0
As S passes through the point (3,1),
(4)2+02+λ(3+12)=0λ=4

S:x2+y2+2x+2y6=0
Centre (1,1) and r=(1)2+(1)2+6=22
If (a,b) is the point on the circle, then the maximum value of a is 1+22.
So, the other root of the quadratic equation will be 122.
Hence, required quadratic equation is
x2(2)x+(1)2(22)2=0x2+2x7=0

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