Question:

For any complex number w=c+w=c+ id, let arg(w)(π,π]\arg (w) \in(-\pi, \pi], where i=1i=\sqrt{-1} Let α\alpha and β\beta be real numbers such that for all complex numbers z=x+z=x+ iy satisfying arg(z+αz+β)=π4\arg \left(\frac{z+\alpha}{z+\beta}\right)=\frac{\pi}{4}, then ordered pair (x,y)(x, y) lies on the circle
x2+y2+5x3y+4=0x^{2}+y^{2}+5 x-3 y+4=0
Then which of the following statements is(are) TRUE?

Updated On: May 23, 2024
  • α=1\alpha=-1
  • αβ=4\alpha \beta=4
  • αβ=4\alpha \beta=-4
  • β=4\beta=4
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The Correct Option is B, D

Solution and Explanation

Circle x2 + y2 + 5x – 3y + 4 = 0 cuts the real axis (x-axis) at (-4, 0), (-1, 0)

Clearly α = 1 and β = 4

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