If the equation- x 2- bx -45-0 b - R has conjugate complex roots and they satisfy - z -1-2 -10- then-A- b2-b-12B- b 2- b -30C- b2-b-72D- b2-b-42

Given x^2 + bx + 45 =0, b ∈ℝ Let roots of the equation be p ± iq Then, sum of roots = 2p = b Product of roots = p^2 + q^2 =45 As p ± iq lie on |z+1| = 2√(10), ...

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Standard XII

Mathematics

Modulus of a Complex Number

If the equati...

Question

If the equation, $x^{2} + b x + 45 = 0 (b \in R)$ has conjugate complex roots and they satisfy $| z + 1 | = 2 \sqrt{10}$ , then:

b^{2} + b = 12

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b^{2} - b = 42

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b^{2} - b = 30

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b^{2} + b = 72

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Solution

The correct option is C $b^{2} - b = 30$
Given $x^{2} + b x + 45 = 0, b \in R$
Let roots of the equation be $p \pm i q$
Then, sum of roots $= 2 p = - b$
Product of roots $= p^{2} + q^{2} = 45$

As $p \pm i q$ lie on $| z + 1 | = 2 \sqrt{10}$ , we get
$(p + 1)^{2} + q^{2} = 40$
$\Rightarrow p^{2} + q^{2} + 2 p + 1 = 40$
$\Rightarrow 45 - b + 1 = 40$
$\Rightarrow b = 6$
$\Rightarrow b^{2} - b = 30.$

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If the equation, x2+bx+45=0 (b∈R) has conjugate complex roots and they satisfy |z+1|=2√10, then:

The correct option is C b2−b=30Given x2+bx+45=0,b∈R Let roots of the equation be p±iq Then, sum of roots =2p=−b Product of roots =p2+q2=45 As p±iq lie on |z+1|=2√10, we get (p+1)2+q2=40 ⇒p2+q2+2p+1=40 ⇒45−b+1=40 ⇒b=6 ⇒b2−b=30.

If the equation, $x^{2} + b x + 45 = 0 (b \in R)$ has conjugate complex roots and they satisfy $| z + 1 | = 2 \sqrt{10}$ , then: