यदि ` (If) 3 sqrt ( a+ ib ) = x + iy `, तो सिद्ध कीजिये कि ` ( a )/ ( x ) + ( b ) /(y ) = 4 ( x ^ 2 - y ^ 2 ) `.

Explanation

प्रश्न से, ` 3sqrt ( a + ib ) = x +iy `
` rArr a + ib = (x + iy ) ^ 3 = x ^ 3 +3x^(2) .iy+ +3xt^(2) y^(2) + i^(3) y^(3) `
` " " =x^(3) + 3x^(2) yi -3xy^(2) -iy^(3)" "[ because i^(2) =-1,i^(3) -i]`
` " = x^(3) -3xy^(2)+ i( 3x^(2) y- y^(3) `
Real तथा imaginary parts को बराबर करने पर ltBrgt ` {:( a = x ^3 - 3xy ^ 2 ) , ( "तथा " b = 3x ^ 2 y - y^ 3 ) :}}" "{:( therefore ( a ) / (x ) = x ^ 2 - 3y ^ 2 ) ,( "तथा " ( b ) /( y ) = 3x^ 2 - y ^ 2 ) :}`
` therefore " "(a)/(x) + (b)/(y) =4x^(2) -4y ^(2) =4( x^(2) -y^(2))`