For any complex number w=c+id, let argw∈(-π,π], where i=-1. Let α and β be real numbers such that for all complex numbers z=x+iy satisfying argz+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?

JEE Advanced · Mathematics · 3. Complex Numbers

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?

A $α = - 1$
B $α β = 4$
C $α β = - 4$
D $β = 4$

Verified Solution

Answer & Solution

Correct Answer

(D) $β = 4$

Step-by-step Solution

Detailed explanation

\arg (\frac{z + α}{z + β}) = \frac{π}{4}

represents an arc of a circle and

(- α, 0) & (- β, 0)

subtends an angle

\frac{π}{4}

z

Given,

x^{2} + y^{2} + 5 x - 3 y + 4 = 0

Centre

(\frac{- 5}{2}, \frac{3}{2}), r = \sqrt{\frac{25}{4} + \frac{9}{4} - 4} = \frac{3}{\sqrt{2}}

Angle made by

(- α, 0) & (- β, 0)

at the centre

C

\frac{π}{2}

Using rotation of complex numbers, we get

\frac{- α - (- \frac{5}{2} + \frac{3}{2} i)}{- β - (- \frac{5}{2} + \frac{3}{2} i)} = 1 \cdot e^{i \frac{π}{2}} = i

- α + \frac{5}{2} - \frac{3}{2} i = - i β + \frac{5}{2} i + \frac{3}{2}

α - i β = 1 - 4 i

∵ α, β \in R

\Rightarrow α = 1, β = 4

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Explore more questions on app

From JEE Advanced

Explore more questions on app

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

Step-by-step Solution