If both the distinct roots of the equation x 2- ax - b -0 a - b - R are lying in between -2 and 2 - thenA- - a -2-b-2C- - a - b -2-2

The correct options are A |a| lt;2 b2 C |a| lt;4 nbsp; nbsp; D |a| gt; b2 2 nbsp; nbsp; Required conditions are (1) f( 2) gt;0, (2) f(2) gt ...

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Byju's Answer

Standard XII

Mathematics

Location of Roots

If both the d...

Question

If both the distinct roots of the equation $x^{2} - a x - b = 0 (a, b \in R)$ are lying in between $- 2$ and $2$ , then

| a | < 2 - \frac{b}{2}

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| a | > 2 - \frac{b}{2}

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| a | < 4

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| a | > \frac{b}{2} - 2

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Solution

The correct options are
A $| a | < 2 - \frac{b}{2}$
C $| a | < 4$
D $| a | > \frac{b}{2} - 2$

Required conditions are
$(1) f (- 2) > 0, (2) f (2) > 0, (3) - 2 < - \frac{B}{2 A} < 2, (4) D > 0$

$f (- 2) > 0$
$\Rightarrow 4 + 2 a - b > 0$
$\Rightarrow a > \frac{b}{2} - 2$
$\Rightarrow - (2 - \frac{b}{2}) < a . . . (i)$

$f (2) > 0$
$\Rightarrow 4 - 2 a - b > 0$
$\Rightarrow a < 2 - \frac{b}{2} . . . (i i)$

From $(i)$ and $(i i),$
$- (2 - \frac{b}{2}) < a < 2 - \frac{b}{2}$
$\Rightarrow | a | < 2 - \frac{b}{2}$

Since, $| a | \geq 0$
$\Rightarrow 2 - \frac{b}{2} > 0$
$\Rightarrow \frac{b}{2} - 2 < 0 \leq | a |$
$\Rightarrow | a | > \frac{b}{2} - 2$

$- 2 < - \frac{B}{2 A} < 2$
$\Rightarrow - 2 < \frac{a}{2} < 2$
$\Rightarrow - 4 < a < 4$
$\Rightarrow a \in (- 4, 4)$
$\Rightarrow | a | < 4$

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