QJEEAdv21Maef86: JEE Advanced 2021 Mathematics - Class 11 JEEAdv21Maef86 Solution: For any complex number w = c + id, let arg ⁡ ( w ) ∈ ( − π , π…

Question

Verified Solution • 2025–26 Session moderate

For any complex number w = c + id, 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 + iy 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?

Accepted Answer

Circle

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

cuts the real axis (X-axis) at (

-

4, 0), (

-

1, 0).

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

implies z is on arc and (

-

α

, 0) and (

-

β

, 0) subtend

\frac{π}{4}

on z.

So,

α

= 1 and

β

= 4

Hence,

α

β

= 1

\times

4 = 4 and

β

= 4

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