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प्रश्न
If (x1 + iy1)(x2 + iy2)(x3 + iy3) ... (xn + iyn) = a + ib, show that `(x_1^2 + y_1^2)(x_2^2 + y_2^2)(x_3^2 + y_3^2) ... (x_"n"^2 + y_"n"^2)` = a2 + b2
उत्तर
Let (x1 + iy1) (x2 + iy2) (x3 + iy3) …… (xn + iyn) = a + ib
Taking modulus on both sides
|(x1 + iy1) (x2 + iy2) (x3 + iy3) …… (xn + iyn)| = |a + ib|
|x1 + iy1| |x2 + iy2| |x3 + iy3| …… |xn + iyn| = |a + ib|
`sqrt(x_1^2 + y_1^2) sqrt(x_2^2 + y_2^2) sqrt(x_3^2 + y_3^2) ... sqrt(x_"n"^2 + y_"n"^2) = sqrt("a"^2 + "b"^2)`
Squaring on both sides
`(x_1^2 + y_1^2)(x_2^2 + y_2^2)(x_3^2 + y_3^2) ... (x_"n"^2 + y_"n"^2)` = a2 + b2
Hence proved.
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