<p>ABC is a triangular park with AB = AC = 100 m. A vertical tower is situated at the mid-point of BC. If the angles of elevation of the top of the tower at A and B are \(\cot^{-1}(3\sqrt{2})\) and \(\csc^{-1}(2\sqrt{2})\) respectively, then the height of the tower (in m) is (JEE Main 2019)</p>
Step-by-Step Solution
Key Concept: Use the geometry of an isosceles triangle with a tower at the midpoint of the base. Convert inverse trigonometric functions to their standard forms, then apply angle of elevation relationships from two different points.
<p><strong>Step 1: Set up the coordinate system.</strong></p><p>Let D be the midpoint of BC where the tower is situated. Let h be the height of the tower. Since AB = AC = 100 m and triangle ABC is isosceles, let AD ⊥ BC (the median from A is also the altitude).</p><p>Place D at origin, with BC along the x-axis and the tower along the y-axis. Then A is at distance AD from D, B is at distance BD from D on the negative x-axis.</p><p><strong>Step 2: Convert the angles of elevation.</strong></p><p>Angle of elevation at A: α = cot⁻¹(3√2)</p><p>If cot(α) = 3√2, then tan(α) = 1/(3√2) = √2/6</p><p>Therefore: tan(α) = h/AD, so h/AD = √2/6 ... (1)</p><p>Angle of elevation at B: β = csc⁻¹(2√2)</p><p>If csc(β) = 2√2, then sin(β) = 1/(2√2) = √2/4</p><p>From sin²(β) + cos²(β) = 1: cos²(β) = 1 - 1/8 = 7/8, so cos(β) = √(7/8)</p><p>Therefore: tan(β) = sin(β)/cos(β) = (√2/4)/(√14/4) = √2/√14 = 1/√7</p><p>From point B: tan(β) = h/BD, so h/BD = 1/√7 ... (2)</p><p><strong>Step 3: Use the constraint AB = 100.</strong></p><p>In right triangle ABD: AB² = AD² + BD²</p><p>100² = AD² + BD²</p><p>10000 = AD² + BD² ... (3)</p><p><strong>Step 4: Express AD and BD in terms of h.</strong></p><p>From equation (1): AD = 6h/√2 = 3√2h</p><p>From equation (2): BD = √7h</p><p><strong>Step 5: Substitute into equation (3).</strong></p><p>(3√2h)² + (√7h)² = 10000</p><p>18h² + 7h² = 10000</p><p>25h² = 10000</p><p>h² = 400</p><p>h = 20 m</p><p><strong>Verification:</strong> AD = 3√2(20) = 60√2 m, BD = √7(20) = 20√7 m</p><p>AD² + BD² = 7200 + 2800 = 10000 ✓</p><p><strong>∴ Answer: A</strong></p>
Correct Answer: A