Let - - be real and z be a complex number- If z 2-2z - - - 0 has two distinct roots- on the line Rez - 1 then it is necessary that

Let P and Q be two points denoting the complex numbers $α a n d β$ respectively on the complex plane. Which of the following equations can represent the equation of the circle passing through P and Q with least possible area ?

Q. If

| 2 z - 1 | = | z - 2 |

and

z_{1}, z_{2}, z_{3}

are complex numbers such that

| z_{1} - α | < α, | z_{2} - β | < β

, then

| \frac{z_{1} + z_{2}}{α + β} |

Q. Let

z

be an imaginary complex number satisfying

| z - 1 | = 1.

α = 2 z,

β = 2 α

and

γ = 2 β,

then the value of

| z |^{2} + | α |^{2} + | β |^{2} + | γ |^{2} + | z - 2 |^{2} + | α - 4 |^{2} + | β - 8 |^{2} + | γ - 16 |^{2}

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Let α, β be real and z be a complex number. If z2+2z + β = 0 has two distinct roots, on the line Re(z) = 1 then it is necessary that

The correct option is D β ϵ[1, ∞)Let roots be p + iq and p - iq, p,q ϵ RSome roots lie on the line Re(z) = 1 ⇒ p = 1 product of roots = p2 + q2 = p = 1 + q2⇒ β ϵ [1,∞) (q ≠ 0 ∵ roots are unequal)

Let $α$ , $β$ be real and z be a complex number. If z $^{2}$ +2z + $β$ = 0 has two distinct roots, on the line Re(z) = 1 then it is necessary that

The correct option is D $β$ $ϵ$ [1, $\infty$ )
Let roots be p + iq and p - iq, p,q $ϵ$ R
Some roots lie on the line Re(z) = 1
$\Rightarrow$ p = 1 product of roots = p $^{2}$ + q $^{2}$ = p = 1 + q $^{2}$
$\Rightarrow$ $β$ $ϵ$ $[1, \infty)$ (q $\neq$ 0 $∵$ roots are unequal)