# If the real part of z¯+2z¯-1 is 4, then show that the locus of the point representing z in the complex plane is a circle. - Mathematics

Sum

If the real part of (barz + 2)/(barz - 1) is 4, then show that the locus of the point representing z in the complex plane is a circle.

#### Solution

Let z = x + iy

∴ barz = x – iy

So(barz + 2)/(barz - 1) = (x - iy + 2)/(x - iy - 1)

= ((x + 2) - iy)/((x - 1) - iy)

= ((x + 2) - iy)/((x - 1) - iy) xx ((x - 1) + iy)/((x - 1) + iy)

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

= (x^2 + 2x - x - 2 + (x + 2 - x + 1)yi + y^2)/((x - 1)^2 + y^2)

= (x^2 + y^2 + x - 2)/((x - 1)^2 + y^2) + (3y)/((x - 1)^2 + y^2)i

Real part = 4

∴ (x^2 + y^2 + x - 2)/((x - 1)^2 + y^2) = 4

⇒ x2 + y2 + x – 2 = 4[(x – 1)2 + y2]

⇒ x2 + y2 + x – 2 = 4[x2 + 1 – 2x + y2]

⇒ x2 + y2 + x – 2 = 4x2 + 4 – 8x + 4y2

⇒ x2 – 4x2 + y2 – 4y2 + x + 8x – 2 – 4 = 0

⇒ – 3x2 – 3y2 + 9x – 6 = 0

⇒ x2 + y2 – 3x + 2 = 0

Which represents a circle.

Hence, z lies on a circle.

Concept: Algebraic Operations of Complex Numbers
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#### APPEARS IN

NCERT Mathematics Exemplar Class 11
Chapter 5 Complex Numbers and Quadratic Equations
Exercise | Q 9 | Page 91

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