Trigonometry & Inverse Trigonometry
Trigonometry
star_batch_jee_advanced_2025
Grade 11
Question:
Two rays are drawn through a point $A$ at an angle of $30°$. A point $B$ is taken on one of them at a distance $a$ from the point $A$. A perpendicular is drawn from the point $B$ to the other ray and another perpendicular is drawn from its foot to $AB$ to meet $AB$ at another point from where the similar process is repeated indefinitely. The length of the resulting infinite polygon line is:
a(2+√3)
a(2-√3)
a
None of these
Step-by-Step Solution
Key Concept: Infinite geometric series with first term $a\sin 30°$ and ratio $\cos 30°$ converges to $a(2 + \sqrt{3})$.
Given perpendiculars $B_1, B_2, B_3, ...$ with feet $BB_1 = a\sin 30°$ and $AB_1 = a\cos 30°$. Each subsequent segment follows the pattern $B_1B_2 = AB_1 \sin 30° = a\cos 30° \sin 30°$ and $AB_2 = AB_1 \cos 30° = a\cos^2 30°$. The total length of the infinite polygonal line is the geometric series $a\sin 30° + a\sin 30°\cos 30° + a\sin 30°\cos^2 30° + ... = \frac{a}{2}\left(1 + \sqrt{3} + \left(\frac{\sqrt{3}}{2}\right)^2 + ...\right) = a(2 + \sqrt{3})$.
Correct Answer: 2