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Properties of Conjugate of a Complex Number

For the equat...

Question

For the equation $a x^{2} + b x + c = 0,$ $a, b$ and $c$ are real,
Statement 1: If the equation $a x^{2} + b x + c = 0, 0 < a < b < c$ , has non-real complex roots $z_{1}$ and $z_{2}$ , then $| z_{1} | > 1, | z_{2} | > 1$ .
Statement 2: Complex roots always occur in conjugate pairs.

A

Both the statements are true, and Statement 2 is the correct explanation for Statement 1.

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B

Both the statements are true, but Statement 2 is not the correct explanation for Statement 1.

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C

Statement 1 is true and Statement 2 is false.

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D

Statement 1 is false and Statement 2 is true.

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Solution

The correct option is A Both the statements are true, and Statement 2 is the correct explanation for Statement 1.
If roots of $a x^{2} + b x + c = 0$ , 0 < a < b < c, are nonreal, then they will be the conjugate of each other. Hence,
$z_{2} = {¯ ¯ ¯ z}_{1} \Rightarrow | z_{1} | = | z_{2} |$
Now, $z_{1} z_{2} = \frac{c}{a} > 1 \Rightarrow | z_{1} |^{2} > 1$
$\Rightarrow | z_{1} | > 1$
$\Rightarrow | z_{2} | > 1$

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