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प्रश्न
Show that `|(z - 2)/(z - 3)|` = 2 represents a circle. Find its centre and radius.
उत्तर
Given that: `|(z - 2)/(z - 3)|` = 2
Let z = x + iy
∴ `|(x + iy - 2)/(x + iy - 3)|` = 2
⇒ `|((x - 2) + iy)/((x - 3) + iy)|` = 2 ......`[because |a/b| = |a|/|b|]`
⇒ `|(x - 2) + iy| = 2|(x - 3) + iy|`
⇒ `sqrt((x - 2)^2 + y^2) = 2sqrt((x - 3)^2 + y^2)`
Squaring both sides, we get
(x – 2)2 + y2 = 4[(x – 3)2 + y2]
⇒ x2 + 4 – 4x + y2 = 4[x2 + 9 – 6x + y2]
⇒ x2 + y2 – 4x + 4 = 4x2 + 4y2 – 24x + 36
⇒ 3x2 + 3y2 – 20x + 32 = 0
⇒ `x^2 + y^2 - 20/3 x + 32/3` = 0
Here g = `(-10)/3`, f = 0
r = `sqrt(g^2 + f^2 - "c")`
= `sqrt(100/9 + 0 - 32/3)`
= `sqrt(100/9 - 32/3)`
= `sqrt((100 - 96)/9)`
= `sqrt(4/9)`
= `2/3`
Hence, the required equation of the circle is `x^2 + y^2 - 20/3 x + 32/3` = 0
Centre = (–g, –f) = `(10/3, 0)` and r = `2/3`.
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