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Question

Let $alpha comma blank beta element of R$ . If $alpha comma blank beta to the power of 2 end exponent$ be the roots of quadratic equation $x to the power of 2 end exponent minus p x plus 1 equals 0$ and $alpha to the power of 2 end exponent comma beta$ be the roots of quadratic equation $x to the power of 2 end exponent minus q x plus 8 equals 0$ , then the value of ‘ $r$ ’ if $fraction numerator r over denominator 8 end fraction$ be arithmetic mean of $p$ and $q$ is

$\frac{83}{2}$
83
$\frac{83}{8}$
$\frac{83}{4}$

The correct answer is: 83

For the equation $x to the power of 2 end exponent minus p x plus 1 equals 0$ ,
The product of roots, $alpha beta to the power of 2 end exponent equals 1$
And for the equation $x to the power of 2 end exponent minus q x plus 8 equals 0$ ,
The product of roots $alpha to the power of 2 end exponent beta equals 8$
Hence, $open parentheses alpha beta to the power of 2 end exponent close parentheses open parentheses alpha to the power of 2 end exponent beta close parentheses equals 8$
$rightwards double arrow blank alpha to the power of 3 end exponent beta to the power of 3 end exponent equals 8 blank rightwards double arrow blank alpha beta equals 2$
$therefore$ From $alpha beta to the power of 2 end exponent equals 1 comma blank$ we have $blank beta equals fraction numerator 1 over denominator 2 end fraction$ and from $blank alpha to the power of 2 end exponent bullet beta equals 8 comma blank$ we have $blank alpha equals 4$
Hence, from sum of roots $equals negative fraction numerator b over denominator a end fraction$ , we have
$p equals alpha plus beta to the power of 2 end exponent equals 4 plus fraction numerator 1 over denominator 4 end fraction equals fraction numerator 17 over denominator 4 end fraction$ and $q equals alpha to the power of 2 end exponent plus beta equals 16 plus fraction numerator 1 over denominator 2 end fraction equals fraction numerator 33 over denominator 2 end fraction$
$fraction numerator r over denominator 8 end fraction$ is arithmetic mean of $p$ and $q$
$therefore blank fraction numerator r over denominator 8 end fraction equals fraction numerator p plus q over denominator 2 end fraction$
$rightwards double arrow r equals 4 open parentheses p plus q close parentheses equals 4 open parentheses fraction numerator 17 over denominator 4 end fraction plus fraction numerator 33 over denominator 2 end fraction close parentheses equals 17 plus 66 equals 83$

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