# If the imaginary part of 2z+1iz+1 is –2, then show that the locus of the point representing z in the argand plane is a straight line. - Mathematics

Sum

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.

#### Solution

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.

Concept: Argand Plane and Polar Representation
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#### APPEARS IN

NCERT Mathematics Exemplar Class 11
Chapter 5 Complex Numbers and Quadratic Equations
Solved Examples | Q 4 | Page 79

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