Question

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

• A. $b^2+b=12$
• B. $b^2-b=42$
• C. $b^2-b=30$
• D. $b^2+b=72$

Answer 1

The correct option is C

$b^2-b=30$

Step 1: Explanation for correct option

option (C): Given $x^2+bx+45=0(b\in \mathrm{ℝ})$ has conjugate complex roots

Let roots of the equation be$p\pm iq$

We know that if $\alpha ,\beta$ are the roots of the quadratic equation $p{x}^2+qx+r=0$ then

$\alpha +\beta =-\frac{q}{p}$

$\alpha \beta =\frac{r}{p}$

Therefore , sum of roots $=2p=-b$

Product of roots $=\left(p-iq\right)\left(p+iq\right)=p^2+q^2=45$

As$p\pm iq$ lies on $|z+1|=2\sqrt{10}$, we get

$$\begin{gathered} \therefore {(p+1)}^2+q^2=40 \\ \Rightarrow p^2+q^2+2p+1=40 \\ \Rightarrow 45-b+1=40 \\ \Rightarrow b=6 \end{gathered}$$

$\therefore b^2-b=6^2-6=30$

So Option (C) is correct

Step 3: Explanation for incorrect options

option (A), option (D):

$$\begin{gathered} b^2+b \\ =6^2+6 \\ =42 \end{gathered}$$

So Option (A) and Option (D) are incorrect

option (B):

$$\begin{gathered} b^2-b \\ =6^2-6 \\ =30\ne 42 \end{gathered}$$

So Option (B) is incorrect

Hence, Option (C) is correct i.e. $b^2-b=30$