Let z =x+ i y be a complex number, where x and y are integers and i = -1 . Then the area of the rectangle whose vertices are the roots of the equation z^3+ z z^3=350 is

Step-by-step solution

aligned & Given, z ( z )^3+ z z ^3=350 & z z ( z )^2+ z zz ^2=350 aligned aligned & |z|^2 ( z )^2+z^2 =350 [ z =|z|^2 ] & |z|^2 [(x- i y)^2+(x+ i y)^2 ]=350 & 2 (x^2+y^2 ) (x^2-y^2 )=350 & 2 (x^4-y^4 )=350 & x^4-y^4=175 & x^4=256, y^4=81 [ x, y are integers ] & x^2=16, y^2=9 & x= 4, y= 3 aligned vertices of the rectangle are (4,3) ;(-4,3) , (-4,-3) and (4,-3) . aligned Required area & = length breadth & =8 6 & =48 sq. units aligned