Question

If the imaginary part of `(2z + 1)/(iz + 1)` is –2, then show that the locus of the point representing z in the argand plane is a straight line.

Answer 1

Let z = x + iy,

Then `(2z + 1)/(iz + 1) = (2(x + iy) + 1)/(i(x + iy) + 1)`

= `((2x + 1) + i2y)/((1 - y) + ix)`

= `({(2x + 1) + i2y})/({(1 - y) + ix}) xx ({(1 - y) - ix})/({(1 - y) - ix})`

= `((2x + 1 - y) + i(2y - 2y^2 - 2x^2 - x))/(1 + y^2 - 2y + x^2)`

Thus `"Im"((2z + 1)/(iz + 1)) = (2y - 2y^2 - 2x^2 - x)/(1 + y^2 - 2y + x^2)`

But `"Im"((2z + 1)/(iz + 1))` = –2  .....(Given)

So `(2y - 2y^2 - 2x^2 - x)/(1 + y^2 - 2y + x^2)` = –2

⇒ 2y – 2y² – 2x² – x = –2 – 2y² + 4y – 2x²

i.e., x + 2y – 2 = 0, which is the equation of a line.