For any complex number w = c + i d , let arg w ∈ ( - π , π ] , where i = - 1 . Let α and β be real numbers such that for all complex numbers z = x + i y satisfying arg z + α z + β = 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?
D. β = 4
arg z + α z + β = π 4 represents an arc of a circle and - α , 0   &   - β , 0 subtends an angle π 4 on z Given, x 2 + y 2 + 5 x - 3 y + 4 = 0 Centre - 5 2 , 3 2 ,   r = 25 4 + 9 4 - 4 = 3 2 Angle made by ( - α , 0 )   &   ( - β , 0 ) at the centre C is π 2 Using rotation of complex numbers, we get - α - - 5 2 + 3 2 i - β - - 5 2 + 3 2 i = 1 · e i π 2 = i - α + 5 2 - 3 2 i = - i β + 5 2 i + 3 2 α - i β = 1 - 4 i ∵ α ,   β ∈ R ⇒ α = 1 ,   β = 4
Related: Mathematics — Complex Number · All PYQ Banks