1.

यदि `| z - 2 | = 2 | z - 1|`, जहाँ z एक सम्मिश्र संख्या है तो सिद्ध कीजिये कि ` |z|^ 2= ( 4 ) / ( 3) Re ( z ) `

Answer» माना कि ` z = x + i y , ` तो प्रश्न से ` | z - 2 | = 2 | z - 1| `
` rArr |x + iy - 2 | = 2 | x + i y - 1| rArr | ( x - 2 ) + iy | = 2| ( x - 1 ) + iy | `
` rArr sqrt (( x - 2 ) ^ 2 + y ^ 2 ) = 2 sqrt (( x - 1 ) ^ 2 + y^ 2 ) rArr ( x- 2 )^ 2 + y^ 2 = 4[ ( x - 1 ) ^ 2 + y ^ 2 ] `
` rArr x^ 2 - 4x + 4 + y^ 2 = 4 ( x ^ 2 - 2 x + 1 + y ^ 2 ) `
` rArr x ^2 + y^ 2 - 4x + 4 = 4x ^ 2 - 8x + 4 + 4y^ 2 `
` rArr 3 ( x ^ 2 + y^ 2 ) = 4x rArr x ^ 2 + y^2 = ( 4 ) / ( 3 ) x rArr |z|^ 2 = ( 4 ) / ( 3 ) Re (z) " "[ because x = Re ( z ) ] `


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