Q.15
The Cartesian coordinates (x, y) of a point A with polar coordinates (4, π/4) is
- (A) (√3, 2√2)
- (B) (2, 2√3)
- (C) (2√2, √3)
- (D) (2√2, 2√2)
Polar coordinates (r, θ) convert to Cartesian coordinates (x, y) using the formulas x = r cos θ and y = r sin θ. For the point (4, π/4), these yield the correct Cartesian coordinates matching option (D).
Correct Answer
The Cartesian coordinates of the point A with polar coordinates (4, π/4) are (2√2, 2√2).
Apply the conversion formulas directly:
x = 4 cos(π/4) = 4 × √2/2 = 2√2
y = 4 sin(π/4) = 4 × √2/2 = 2√2
Option Analysis
| Option | Coordinates | Why Incorrect/Correct |
|---|---|---|
| (A) | (3√2 , 2√2) | Uses cos(π/6)=√3/2 incorrectly; y matches but x does not. |
| (B) | (2, 2√3) | Suggests r = 4 at θ = π/3 where sin(π/3) = √3/2; wrong angle. |
| (C) | (2√2 , √3) | x matches π/4 but y implies sin(π/6)=1/2; inconsistent trig values. |
| (D) | (2√2 , 2√2) | Exact match: both cos(π/4) and sin(π/4) equal √2/2. |
