Question

If cartesian co-ordinates of a point are `(1, -sqrt3)`, then its polar co-ordinates are ______ 

पर्याय

• `(1, (5pi)/3)`

• `(2, (5pi)/3)`

• `(1, (3pi)/4)`

• `(2, (3pi)/4)`

Answer 1

If cartesian co-ordinates of a point are `(1, -sqrt3)`, then its polar co-ordinates are `underline((2, (5pi)/3))`.

Explanation:

(x, y) ≡ `(1, -sqrt3)`

x = r cos θ and y= r sin θ

⇒ 1 = r cosθ, `-sqrt3` = r sinθ

Now, `r = sqrt(x^2 + y^2) = sqrt(1 + 3) = 2`

and tanθ = `(r sintheta)/(r costheta) = -sqrt3 = -tan  pi/3`

∴ tanθ = `tan(pi - pi/3) = tan  (2pi)/3` ........[∵ tan(π - θ) = -tanθ]

⇒ tanθ = `tan  (2pi)/3`

⇒ `theta = npi + (2pi)/3`, n ∈ Z

∴ `theta = (2pi)/3, (5pi)/3` ......[0 ≤ θ < 2π]

But the given point lies in the 4^th quadrant.

∴ `theta = (5pi)/3`

∴ The required polar co-ordinates are `(2, (5pi)/3)`