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Question
Find the locus of a complex number, z = x + iy, satisfying the relation `|[ z -3i}/{z +3i]| ≤ sqrt2 `. Illustrate the locus of z in the Argand plane.
Solution
`|[ z -3i]/[z +3i]| ≤ sqrt2 `
⇒ `|[x + iy - 3i]/[x +iy +3i]| ≤ sqrt2 `
⇒ `|x + i (y - 3)| ≤ sqrt2 |x + i (y + 3)|`
⇒ `sqrt(x^2 + (y - 3)^2) ≤ sqrt2 sqrt(x^2 + (y + 3)^2)`
⇒ x2 + (y - 3)2 ≤ 2 (x2 + (y + 3)2)
⇒ x2 + y2 + 9 - 6y ≤ 2x2 + 2(y2 + 9 + 6y)
⇒ 2x2 + 2y2 + 18 + 12y - x2 - y2 - 9 + 6y ≥ 0
⇒ x2 + y2 + 9 + 18y ≥ 0
⇒ x2 + y2 + 18y + 9 + 81 ≥ 81
⇒ x2 + (y + 9)2 ≥ 72
⇒ (x - 0)2 + (y - (-9))2 ≥ ( 6√2)2
This represents a circle with centre (0, 9) and radius 6√2 units.
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