If $p, q, r$ & $s$ are real numbers & $p r = 2 (q + s)$ , then which of the following statement is correct about these equations $x^{2} + p x + q = 0, x^{2} + r x + s = 0$

Both equations must have real roots

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Atleast one of the equation must have real roots

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Both equations must have imaginary roots

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Exactly one of the equation must have real roots

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Solution

The correct option is B Atleast one of the equation must have real roots
Given, $x^{2} + p x + q = 0, x^{2} + r x + s = 0$ and $p r = 2 (q + s)$
$D_{1} = p^{2} - 4 q$ & $D_{2} = r^{2} - 4 s$
Add $D_{1} a n d D_{2}$
$D_{1} + D_{2} = p^{2} + r^{2} - 4 (q + s)$
$= p^{2} + r^{2} - 2 p r$ (from given)
$= {(p - r)}^{2} \geq 0$
$\Rightarrow D_{1} + D_{2} \geq 0$ possible only if at least one of $D_{1}$ or $D_{2}$ is $> 0$ .

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If p,q,r & s are real numbers & pr=2(q+s), then which of the following statement is correct about these equations x2+px+q=0,x2+rx+s=0

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If $p, q, r$ & $s$ are real numbers & $p r = 2 (q + s)$ , then which of the following statement is correct about these equations $x^{2} + p x + q = 0, x^{2} + r x + s = 0$