Show the Transformation 𝑤=(2𝑧+3)/(𝑧-4) transform the circle 𝑥²+𝑦²-4𝑥=0 into the straight line 4𝑢+3=0

𝑪𝒐𝒎𝒑𝒍𝒆𝒙 𝑨𝒏𝒂𝒍𝒚𝒔𝒊𝒔 𝑻𝒉𝒆𝒐𝒓𝒆𝒎𝒇𝒐𝒎 𝑪𝒐𝒏𝒇𝒐𝒓𝒂𝒍 𝑻𝒓𝒂𝒔𝒇𝒐𝒎𝒂𝒕𝒊𝒐𝒏 𝑻𝑯𝑬𝑶𝑹𝑬𝑴/ 𝑺𝑻𝑨𝑻𝑬𝑴𝑬𝑵𝑻 / 𝑷𝑹𝑶𝑽𝑬 𝑻𝑯𝑨𝑻 :- Show that the Transformation 𝑤=(2𝑧+3)/(𝑧-4) transform the circle 𝑥²+𝑦²-4𝑥=0 into the straight line 4𝑢+3=0 . 𝑷𝑹𝑶𝑶𝑭 :- Given that 𝑤=(2𝑧+3)/(𝑧-4) ⇒𝑤(𝑧-4)=(2𝑧+3) ⇒𝑤𝑧-4𝑤=2𝑧+3 ⇒𝑤𝑧-4𝑤-2𝑧-3=0 ⇒𝑤𝑧-2𝑧-4𝑤-3=0 ⇒(𝑤-2)𝑧-4𝑤-3=0 ⇒𝑧=(4𝑤+3)/(𝑤-2) ⎞ Its conjugate is ⎬ .....(1) ⇒ 𝑧̅=(4𝑤̅+3)/(𝑤̅-2) ⎠ ∵ the given circle is 𝑥²+𝑦²-4𝑥=0 we know that 𝑧𝑧̅=𝑥²+𝑦² & 𝑧+𝑧̅=2𝑥 ∴ 𝑧𝑧̅-2(𝑧+𝑧̅)=0 Putting from equation (1) {(4𝑤+3)(4𝑤̅+3)}/{(𝑤-2)(𝑤̅-2)}-2[{(4𝑤+3)/(𝑤-2)}+{(4𝑤̅+3)/(𝑤̅-2)}]=0 Multiplying by (𝑤-2)(𝑤̅-2) ⇒(4𝑤+3)(4𝑤̅+3)-2{(4𝑤+3)(𝑤̅-2)}+(4𝑤̅+3)(𝑤-2)}=0 ⇒(16𝑤𝑤̅+12𝑤+12𝑤̅+9)-2(4𝑤𝑤̅-8𝑤+3𝑤̅-6+4𝑤𝑤̅-8𝑤̅+3𝑤-6)=0 ⇒16𝑤𝑤̅+12𝑤+12𝑤̅+9-8𝑤𝑤̅+16𝑤-6𝑤̅+12-8𝑤𝑤̅+16𝑤̅-6𝑤-12=0 ⇒22(𝑤+𝑤̅)+33=0 ⇒2(𝑤+𝑤̅)+3=0 ∵ 𝑤+𝑤̅=2𝑢 ∴2×2𝑢+3=0 ⇒4𝑢+3=0 ------------------------------------------------------------------ Proved...!!! . . . #complexanalysis #csirnet #conformal #isogonal #transformation #mapping #bilineartransformation #mobius #lineartransformation #fractionaltransformation #fixedpoints . 🔥🔥🔥Hurry Up Students for whole detailed Videos with Answer in Description for making notes 🔥🔥🔥 🔥🔥🔥JOIN JOIN JOIN JOIN 🔥🔥🔥 👉    / @abcstudy   🔥🔥🔥All B.Sc., B.Tech , M.Sc. , M.A. Mathematics students join with us A great platform to get knowledge in Mathematics and science... Stay tuned....for detailed videos on whole SYLLABUS with topic wise...and also join us for CSIR-UGC National Eligibility Test (NET) for Junior Research Fellowship and Lecturer-ship COMMON SYLLABUS FOR PART 'B' AND 'C' MATHEMATICAL SCIENCE 🔥🔥🔥 ------------------------------------------------------------------ Channel Link: 🔥🔥🔥    / @abcstudy   Videos links 🔥🔥🔥 👇👇👇👇 ------------------------------------------------------------------ 52)    • Show the Transformation 𝑤=(2𝑧+3)/(𝑧-4) tra...   51)    • Prove the resultant (or product) of two Bi...   50)    • "Bilinear Transformation" as a combination...   49)    • Fixed Points in Bilinear Transformation 𝑧=...   48)    • Critical Points in Bilinear Transformation   47)   • Video   46)    • Find image of 0 ≺ 𝑹𝒆 𝑧 ≺ 𝜋/2 Under conform...   45)    • 𝑤=√2.𝑒ⁱ⁽  ⷫᐟ⁴⁾𝑧+(1-2𝑖) & determine region ...   44)    • Combined Elementary transformation : LINEA...   43)    • Combined Elementary transformation : Rotat...   42)    • Show transformation 𝑤(𝑧+𝑖)²=1 maps interio...   41)    • Elementary Transformation (iv) Inversion 𝑤...   40)    • Elementary Transformation (iii) Magnificat...   39)    • Elementary Transformation (ii) Rotation 𝑤 ...   38)    • Video   37)    • If 𝑓(𝑧) shows Conformal transformation in ...   36)    • Conformal Transformation and Isogonal Tran...   35)    • If 𝑓 (𝑧) is an Analytic function then prov...   34)    • Jacobian of a Transformation (And Inverse ...   33)    • Conformal Transformation : Discribe with e...   32)    • Find radius of convergence of the power se...   ---------------------------‐------------------------------------ ------------------------------------------------------------------ Play List ( Complex Analysis) 👉    • Complex Analysis   ------------------------------------------------------------------ ------------------------------------------------------------------ Kindly join Telegram channel 🔥🔥🔥 👉 https://t.me/abcstudyy ------------------------------------------------------------------