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Question

Let α and β are the roots of the equation ax2+bx+c=0. If 1αα and 1ββ are the roots of px2+qx+r=0 and qp=k+bc, then the value of k is

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Solution

Let f(x)=ax2+bx+c=0 has roots α,β
y=1xx, where x=α,β
y=1x1y+1=1xx=1y+1

The equation whose 1αα and 1ββ is,
f(1x+1)=0a+b(x+1)+c(x+1)2=0cx2+(2c+b)x+(a+b+c)=0
Comparing with
px2+qx+r=0
cp=(2c+b)q=(a+b+c)r=k
Now,
cp=(2c+b)q
qp=(2c+b)cqp=2+bck=2

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