Question

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

Answer 1

Answer:

Let z=x+iy

then

∣

∣

∣

∣

∣

∣

z+3i

z−3i

∣

∣

∣

∣

∣

∣

≤

2

∣

∣

∣

∣

∣

∣

x+iy+3i

x+iy−3i

∣

∣

∣

∣

∣

∣

≤

2

∣

∣

∣

∣

∣

∣

x+i(y+3)

x+i(y−3)

∣

∣

∣

∣

∣

∣

≤

2

x

2

+(y−3)

2

≤

2

x

2

+(y+3)

2

On squaring on both sides, we get

x

2

+(y−3)

2

≤2(x

2

+(y+3)

2

)

x

2

+y

2

+9−6y≤2x

2

+2y

2

+18+12y

x

2

+y

2

+18y+9≥0

A circle with centre (0,−9) and radius 6

2