The distance between the two points A and A' which lie on y = 2 such that both the line segments AB and A' B (where B is the point (2, 3)) subtend angle π/4 at the origin, is equal to :

Solution

Maths sir · 4 steps

Turn the angle into slopes: two lines through O at 45° to OB, then cut them with y = 2.

1. Angle becomes slope. The segment AB subtends π/4 at O, so the lines OA and OB meet at 45°. Line OB has slope 3/2. Call the slope of OA m.

   |(m - 3/2)/(1 + 3/2m)| = tan π/4 = 1 ⇒ |(2m - 3)/(2 + 3m)| = 1

2. Two slopes. Taking 2m - 3 = 2 + 3m gives m = -5. Taking 2m - 3 = -(2 + 3m) gives 5m = 1, so m = 1/5. These are the two lines OA and OA'.
3. Meet y = 2. On y = mx with y = 2, the point is at x = 2/m. Slope 1/5 gives A(10, 2), and slope -5 gives A'(-2/5, 2).
4. So. Both points are on the same horizontal line, so AA' = 10 - (-2/5) = 52/5. The answer is (c).

The idea to keep. When a segment subtends a given angle at the origin, it is really a question about the angle between two lines through O. Use the tan θ slope formula and keep both signs of the modulus.

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