Two lines zi − z i + 2 = 0 and z ( 1 + i ) + z ( 1 − i ) + 2 = 0 intersect at a point P . There is a complex number α = x + iy at a distance of 2 units from the point P which lies on line z ( 1 + i )

Question

Two lines zi − z i + 2 = 0 and z ( 1 + i ) + z ( 1 − i ) + 2 = 0 intersect at a point P . There is a complex number α = x + iy at a distance of 2 units from the point P which lies on line z ( 1 + i ) + z ( 1 − i ) + 2 = 0 . Find [ ∣ x ∣ ] (where [.] represents greatest integer function).

TestHub · Accepted Answer

Solving the equation of the lines we get $z = - \overline{z} \Rightarrow z = i$ $∣ α - 1∣ = 2; α = 2 e^{i θ} + i$ , put it in the equation of the second line, we get

$cos θ - sin θ = 0$

$α = i \pm 2 e^{\frac{i π}{4}} ∴ x = \pm 2 \Rightarrow [∣ x ∣] = 1$

Mathematics - Complex Number Question with Solution | TestHub

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