Question

If x + iy = `5/(3 + costheta + isintheta)`, then x² + y² is equal to ______ 

Options

• `25/(6 + 10costheta)`

• `25/(10 + 6costheta)`

• `5/(10 - 6costheta)`

• `5/(6 - 10costheta)`

Answer 1

If x + iy = `5/(3 + costheta + isintheta)`, then x² + y² is equal to `underline(25/(10 + 6costheta))`.

Explanation:

x + iy = `5/(3 + costheta + isintheta)`

= `(5(3 + costheta - isintheta))/((3 + costheta)^2 + sin^2theta)`

= `(15 + 5costheta - 5isintheta)/(9 + cos^2theta + 6costheta + sin^2theta)`

= `[(15 + 5costheta)/(10 + 6costheta)] + i[(-5sintheta)/(10 + 6costheta)]`

⇒ x = `(5(3 + costheta))/(10 + 6costheta), y = (-5sintheta)/(10 + 6costheta)`

∴ `x^2 + y^2`

= `25/(10 + 6costheta)^2 [9 + cos^2theta + 6costheta + sin^2theta]`

= `25/(10 + 6costheta)`