If under root of (a+i b)/(c+i d)=x+i y , Prove (a^2+b^2)/(c^2+d^2)=(x^2+y^2)^2

Answer

N/a

Explanation

Given : `x- iy=sqrt((a-ib)/(c -i d))" "...(1)`
` rArr" "x+iy=sqrt ((a+ ib)/(c+i d))" "...(2) `
then from equations (1) and (2),
`" "(x-iy)(x+iy)=sqrt((a-ib)/(c-i d))xxsqrt((a+ ib)/(c+i d))`
or `" "x^(2)-i^(2)y^(2)=sqrt(( (a-ib) (a+ ib))/((c-i d)(c+i d)))`
or `" "x^(2)-i^(2)y^(2)=sqrt( ( a^(2)-i ^(2)b^(2))/(c^(2)- i^(2)d^(2)))`
or `" "x^(2)-(-1) y^(2)=sqrt((a^(2)-(-1)b^(2))/(c^(2)-(-1)d^(2)))`
or `" "x^(2)+y^(2)=sqrt((a^(2)+b^(2))/(c^(2)+ d^(2)))`
Therefore `" "(x^(2) +y^(2))^(2)=(a^(2)+b^(2))/(c^(2)+d^(2))`