For any complex number , let , where . Let and be real numbers such that for all complex numbers satisfying the ordered pair lies on the circle . Then which of the following statements is (are) TRUE? [2021]

Question

For any complex number $w = c + i d$ , let $\arg (w) \in (- π, π]$ , where $i = \sqrt{- 1}$ . Let $α$ and $β$ be real numbers such that for all complex numbers $z = x + i y$ satisfying $\arg (\frac{z + α}{z + β}) = \frac{π}{4},$ the ordered pair $(x, y)$ lies on the circle $x^{2} + y^{2} + 5 x - 3 y + 4 = 0$ . Then which of the following statements is (are) TRUE? [2021]

Smart Achievers · Accepted Answer

(2, 4)

Given that $\arg (\frac{z + α}{z + β}) = \arg (z + α) - \arg (z + β) = \frac{π}{4}$ implies $z$ is on arc and $(- α, 0)$ & $(- β, 0)$ subtend $\frac{π}{4}$ on $z$ .

Given that $z$ lies on $x^{2} + y^{2} + 5 x - 3 y + 4 = 0$

So put $y = 0$ ; for value of $α$ and $β$

$x^{2} + 5 x + 4 = 0 \Rightarrow x = - 1, x = - 4$

$∴$ $α = 1, β = 4 .$

Solution

1 $α = - 1$

2 $α β = 4$

3 $α β = - 4$

4 $β = 4$

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Solution

Q. For any complex number w=c+id, let arg(w)∈(-π,π], where i=-1. Let α and β be real numbers such that for all complex numbers z=x+iy satisfying arg(z+αz+β)=π4, the ordered pair (x,y) lies on the circle x2+y2+5x-3y+4=0. Then which of the following statements is (are) TRUE [2021]

1 α=-1

2 αβ=4

3 αβ=-4

4 β=4

Ans. (2, 4) Given that arg(z+αz+β)=arg(z+α)-arg(z+β)=π4 implies z is on arc and (-α,0) & (-β,0) subtend π4 on z. Given that z lies on x2+y2+5x-3y+4=0 So put y=0; for value of α and β x2+5x+4=0 ⇒ x=-1, x=-4 ∴ α=1, β=4.

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1 $α = - 1$

2 $α β = 4$

3 $α β = - 4$

4 $β = 4$