If the equation, x2 + bx + 45 = 0 (b $\in$ R) has conjugate complex roots and they satisfy |z +1| = 2$\sqrt {10}$ , then :
Solution
x<sup>2</sup>
+ bx = 45 = 0 (b $\in$ R)
<br>has roots $\alpha$ + i$\beta$, $\alpha$ – i$\beta$
<br>sum of roots = – b = 2$\alpha$
<br>product of roots = 45 =
$\alpha$<sup>2</sup>
+ $\beta$<sup>2</sup>
<br><br>Let z = x + iy
<br><br>$\therefore$ |x + iy +1| = 2$\sqrt {10}$
<br><br>${\left| {x + iy + 1} \right|^2} = {\left( {2\sqrt {10} } \right)^2}$
<br><br>$\Rightarrow$ (x + 1)<sup>2</sup> + y<sup>2</sup> = 40
<br><br>$\Rightarrow$ ($\alpha$ + 1)<sup>2</sup> + $\beta$<sup>2</sup> = 40
<br><br> [putting real part $\alpha$ in place of x and imaginary part $\beta$ in place of y]
<br><br>$\Rightarrow$ $\alpha$<sup>2</sup> + 2$\alpha$ + 1 + $\beta$<sup>2</sup> = 40
<br><br>$\Rightarrow$ 45 + 2$\alpha$ + 1 = 40
<br><br>$\Rightarrow$ $\alpha$ = -3
<br><br>$\therefore$ -b = 2$\alpha$ = 2$\times$(-3) = -6
<br><br>$\Rightarrow$ b = 6
<br><br>By checking options we found b<sup>2</sup> – b = 30.
About this question
Subject: Mathematics · Chapter: Complex Numbers and Quadratic Equations · Topic: Complex Numbers and Argand Plane
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