If the equation- x2 -bx -45 - 0 b - has conjugate complex roots and they satisfy -z-1- - 2 -10- then-

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), ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XIII

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

No worries! We‘ve got your back. Try BYJU‘S free classes today!

b^{2} - b = 42

No worries! We‘ve got your back. Try BYJU‘S free classes today!

b^{2} - b = 30

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

b^{2} + b = 72

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

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
$(p + 1)^{2} + q^{2} = 40$
$\Rightarrow p^{2} + q^{2} + 2 p + 1 = 40$
$\Rightarrow 45 - b + 1 = 40$
$\Rightarrow b = 6$
$\Rightarrow b^{2} - b = 30.$

Suggest Corrections

Similar questions

Q. 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:

Q. Assertion :If

a^{2} + b^{2} + c^{2} < 0

, then if roots of the equation

a x^{2} + b x + c = 0

are imaginary, then they are not complex conjugates. Reason: Equation

a x^{2} + b x + c = 0

has complex conjugate roots when a, b, c are real.

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.

Q. The equation

z^{10} + (13 z - 1)^{10} = 0

has

5

pairs of complex roots

a_{1}, b_{1}, a_{2}, b_{2}, a_{3}, b_{3}, a_{4}, b_{4}, a_{5}, b_{5}

. If each pair

a_{i}, b_{i}

are complex conjugates, then

Q. Assertion :If the equation

x^{2} + b x + c a = 0

and

x^{2} + c x + a b = 0

have a common root, then their other root will satisfy the equation

x^{2} + a x + b c = 0

Reason: If the equation

x^{2} = b x + c a = 0

and

x^{2} + c x + a b = 0

have a common root, then

a + b + c = 0

Join BYJU'S Learning Program

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: