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In triangle ABC, the segment joining the orthocentre and the circumcentre is parallel to side BC. Statement 1: tan A, tan B and tan C are in arithmetic progression. Statement 2: If tan A, tan B and tan C are in arithmetic progression, then tan A × tan C = 3. Choose the correct option: (a) Statement 1 is true, Statement 2 is true; Statement 2 correctly explains Statement 1 (b) Statement 1 is true, Statement 2 is true; Statement 2 does not correctly explain Statement 1 (c) Statement 1 is false, Statement 2 is true (d) Statement 1 is true, Statement 2 is false

1. a
2. b
3. c
4. d

Correct answer: b

Solution

Statement 1 is true because in a triangle where the segment joining the orthocentre and circumcentre is parallel to a side, the angles satisfy the condition for their tangents to be in arithmetic progression. Statement 2 is also true, but it does not provide a direct explanation for Statement 1, as the relationship between the tangents does not imply the specific product of tangents stated.

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