Complex equation(A JEE Problem)!
Let be a complex number where and are integers. Find the area of the rectangle whose vertices are the roots of the equation:
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1 solution
z z ˉ 3 + z ˉ z 3 z z ˉ ( z ˉ 2 + z 2 ) ( x 2 + y 2 ) ( x 2 − 2 x y i − y 2 + x 2 + 2 x y i − y 2 ) ( x 2 + y 2 ) ( 2 ( x 2 − y 2 ) ) ( x 2 − y 2 ) ( x 2 + y 2 ) = 3 5 0 = 3 5 0 = 3 5 0 = 3 5 0 = 1 7 5
Factorizes 175, we got 175=5^2 × 7. Hence, 175=1×175=5×35=25×7. As x^2 & y^2 are square integers, the only factorization of 175 must be 25×7. (If 1 7 5 = a × b ), then x 2 = 2 a + b and y 2 = 2 a − b )
Hence, x = ± 4 and y = ± 3 , so the area is 8 × 6 = 4 8 .