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Maths-

General

Easy

Question

Let $text end text alpha text end text$ and $text end text beta text end text$ be the roots of the equation $text end text x to the power of 2 end exponent minus p x plus r equals 0 text end text$ and $text end text fraction numerator alpha over denominator 2 end fraction text , end text blank 2 beta text end text$ be the roots of the equation $x to the power of 2 end exponent minus q x plus r equals 0 text . end text$ The value of $text end text text end text r text end text$ is

$\frac{2}{9} (p - q) (2 q - p)$
$\frac{2}{9} (q - p) (2 p - q)$
$\frac{2}{9} (q - 2 p) (2 p - p)$
$\frac{2}{9} (2 p - q) (2 q - p)$

The correct answer is: $fraction numerator 2 over denominator 9 end fraction left parenthesis 2 p minus q right parenthesis left parenthesis 2 q minus p right parenthesis$

$text Since end text alpha text and end text beta text are the roots of equation end text x to the power of 2 end exponent minus p x plus r equals 0 text , end text$ we get
$table row cell alpha plus beta equals p end cell row cell alpha beta equals r end cell end table$
$text Also, since end text alpha divided by 2 text and end text 2 beta text are the roots of equation end text x to the power of 2 end exponent minus q x plus r equals 0 text , end text$
we get
$fraction numerator alpha over denominator 2 end fraction plus 2 beta equals q rightwards double arrow alpha plus 4 beta equals 2 q$
Solving Eqs. (1) and (2), we get
$alpha equals fraction numerator 2 left parenthesis 2 p minus q right parenthesis over denominator 3 end fraction text and end text beta equals fraction numerator 2 q minus p over denominator 3 end fraction$
Substituting a and b in Eq. (2), we get
$r equals fraction numerator 2 over denominator 9 end fraction left parenthesis 2 p minus q right parenthesis left parenthesis 2 q minus p right parenthesis$

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