Let w₁ be the point obtained by the rotation of z₁=5+4 i about the origin through a right angle in the anticlockwise direction, and w₂ be the point obtained by the rotation of z₂=3+5 i about the origin through a right angle in the clockwise direction. Then the principal argument of w₁-w₂ is equal to :

Solution

Maths sir · 4 steps

A right-angle turn is just multiplying by i or -i. Do that, subtract, then check the quadrant.

1. Rotation rule. Rotating a point about the origin by 90° anticlockwise multiplies it by i. Rotating by 90° clockwise multiplies it by -i.
2. Find both points. w₁ = i(5+4i) = 5i + 4i² = -4+5i. Also w₂ = -i(3+5i) = -3i - 5i² = 5-3i.
3. Subtract. w₁ - w₂ = (-4-5) + (5+3)i = -9+8i. The real part is negative and the imaginary part is positive, so the point lies in the second quadrant.

   arg(-9+8i) = π - tan⁻¹ 8/9

4. So. In the second quadrant the principal argument is π minus the acute angle tan⁻¹|y|/|x| = tan⁻¹ 8/9. The answer is (c).

The idea to keep. For a principal argument, first find the acute angle tan⁻¹|y/x|, then let the signs of x and y decide the quadrant. Second quadrant gives π - α, third quadrant gives -π + α.

Answer from the NTA final answer key. Solution by PrepWiser.