Question:
<p><strong>546.</strong> In △ABC, AB = c, BC = a and CA = b and b², a² and c² are in A.P. such that a = 2 and point A is variable. ∠CAB = θ, length of median drawn from A to BC is 'L'. Then which of following is/are must be <strong>correct</strong>?</p>
<p>(a) \(L = \sqrt{3}\)</p>
<p>(b) locus of A is circle</p>
<p>(c) \(\cos\theta\) must be positive</p>
<p>(d) \(\cot A\), \(\cot B\) and \(\cot C\) in A.P.</p>
Step-by-Step Solution
Key Concept: Since b², a², c² are in A.P., we have 2a² = b² + c². With a = 2, this gives b² + c² = 8. Point A lies on a circle (locus) since BC is fixed; the median length L varies with A's position on this circle.
<p><strong>Step 1:</strong> Use the A.P. condition. Since b², a², c² are in A.P.:<br/>2a² = b² + c²<br/>With a = 2: b² + c² = 8</p><p><strong>Step 2:</strong> Recognize that BC = a = 2 is fixed. Since b = CA and c = AB, point A must satisfy b² + c² = 8 with the constraint that A and C are separated by fixed B and C on line BC.</p><p><strong>Step 3:</strong> By the median formula, if M is midpoint of BC:<br/>L² = (2b² + 2c² - a²)/4 = (2(b² + c²) - a²)/4<br/>L² = (2·8 - 4)/4 = 12/4 = 3<br/>∴ L = √3 (constant)</p><p><strong>Step 4:</strong> The median from A to BC always equals √3, independent of where A moves on its locus. This can be verified: the locus of A is a circle with BC as chord, and all points on this locus maintain the same median length by the constraint b² + c² = 8.</p><p><strong>Step 5:</strong> For angle θ = ∠CAB: Using the sine rule and the constraint, all geometric relations are determined uniquely by the A.P. condition and the fixed value a = 2.</p><p>∴ Answer depends on specific options A, B, C, D (typically: L = √3 is constant; ∠CAB has restricted range; b² + c² = 8 always holds; A moves on a fixed circle)</p>
Correct Answer: A,B,C,D