Question

Assertion :Let $b,a,c,p,q$ be real numbers in ascending order.
Suppose $\alpha ,\beta$ are the roots of the equation $x^2+2px+q=0$ and $\alpha ,\frac{1}{\beta }$ are the roots of the equation $a{x}^2+2bx+c=0$, where ${\beta }^2\notin \{-1,0,1\}$

$(p^2-q)(b^2-ac)\ge 0$

Reason: $b\ne paorc\ne qa$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
• B. Both Assertion and Reason are correct, but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.

$x^2+2px+q=0$
Discriminant, $D=(2p{)}^2-(4\left)\right(1\left)\right(q)$
$=4p^2-4q$
$=p^2-q$
As $p$ and $q$ are real numbers with $p>q$
$\Rightarrow D\ge 0$

And in $a{x}^2+2bx+c=0$
Discriminant, $D=(2b{)}^2-4(a\left)\right(c)$
$=4b^2-4ac$
$=b^2-ac$

As $a,b,c$ are real numbers with $b>a>c$
$\Rightarrow D\ge 0$
Hence, Assertion is true

If $b=pa$ and $c=qa$, then two equation becomes equivalent
Which means the roots $\alpha ,\beta$ are roots for two equation, which is not true by given statement
Therefore, Reason is true.