Question

If the ratio of the roots of equation px2+qx+q=0 is a:b, prove that root a/b+root b/a+root q/p

Answer 1

Let α and β are the roots of given equation px² + qx + q = 0
sum of roots = α + β = -q/p --------(1)
product of roots = αβ = q/p -------(2)

Given, ratio of roots = a/b
α/β = a/b --------(3)
multiply equations (2) and (3)
αβ × α/β = q/p × a/b = qa/pb
α² = qa/pb
taking square root both sides,
α = $\bold{\sqrt{\frac{aq}{pb}}}$
put this in equation (2)
β = q/pα = q/p√{aq/pb} = $\bold{\sqrt{\frac{bq}{pa}}}$

Now, put it in equation (1)
α + β = -q/b
$\bold{\sqrt{\frac{aq}{pb}}}+\bold{\sqrt{\frac{bq}{pa}}}=-\bold{\frac{q}{p}}$
$\bold{\sqrt{\frac{q}{p}}}[\bold{\sqrt{\frac{a}{b}+\frac{b}{a}]}}=-\bold{\frac{q}{p}}\\\\\bold{\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}+\sqrt{\frac{q}{p}}}=0$

Hence , proved//

Answer 2

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