The roots of the equation 7x^{2}-6x+1=0 are \tan \alpha and \tan \beta, where 2\alpha and 2\beta are the angles of a triangle. Which one of the following is correct?
- A. The triangle is equilateral
- B. The triangle is isosceles but not right-angled
- C. The triangle is right-angled ✓
- D. The triangle is right-angled isosceles
Correct Answer: C. The triangle is right-angled
Explanation
The sum of roots is \tan\alpha + \tan\beta = \frac{6}{7} and the product is \tan\alpha\tan\beta = \frac{1}{7}. Using the identity \tan(\alpha+\beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta} = \frac{6/7}{1 - 1/7} = \frac{6/7}{6/7} = 1. Thus, \alpha+\beta = 45^{\circ}. The sum of the two angles of the triangle is 2\alpha + 2\beta = 2(45^{\circ}) = 90^{\circ}. Therefore, the third angle is 180^{\circ} - 90^{\circ} = 90^{\circ}, making it a right-angled triangle.