Question

A real value of x satisfies `((3 - 4ix)/(3 + 4ix))` = α – iβ (α, β ∈ R), if α² + β² is equal to

Options

• 1

• – 1

• 2

• – 2

Answer 1

1

Explanation:

Since, `((3 - 4ix)/(3 + 4ix))` = α – iβ (α, β ∈ R)

⇒ `(alpha - ibeta) = ((3 - 4ix)(3 - 4ix))/((3 + 4ix)(3 - 4ix)) = (9 + 16i^2 x^2 - 24ix)/(9 - 16i^2 x^2)`

⇒ `alpha - ibeta = (9 - 16x^2 - 24ix)/(9 + 16x^2)`  .....`[because i^2 = 1]`

⇒ `alpha - ibeta = (9 - 16x^2)/(9 + 16x^2) - (i(24x))/(9 + 16x^2)`

`a^2 + beta^2 = ((9 - 16x^2)/(9 + 16x^2))^2 + ((24x)/(9 + 16x^2))^2`

= `(81 + 256x^4 - 288x^2 + 576x^2)/(9 + 16x^2)^2`

= `(81 + 256x^4 + 288x^2)/(9 + 16x^2)^2`

= `(9 + 16x^2)^2/(9 + 16x^2)^2`

= 1