Question

Roots of the equation x² + bx + 45 = 0, b ∈ R lie on the curve |z + 1| = 2√10, where z is a complex number then

(1) b² + b = 12 

(2) b² – b = 30 

(3) b² – b = 36 

(4) b² + b = 30 

Answer 1

Answer is (2) b² – b = 30

Let z = α ± iβ be roots of the equation 

So 2α = –b and α² + β² = 45, (α + 1)² + β² = 40

So (α + 1)² – α² = – 5

⇒ 2α + 1 = – 5 ⇒ 2α = – 6

so b = 6

hence b² – b = 30