Question

If triangle ABC is a right angled at A and \( \tan \frac{\text{B}}{2}, \tan \frac{\text{C}}{2} \) are roots of the equation \( a x^2 + bx + c = 0, \text{a} \ne 0 \), then

• A. \( \text{a} + c = b \)
• B. \( \text{a} + b = c \)
• C. \( b + c = \text{a} \)
• D. \( a + c = 2b \)

Hint:

If \( \alpha + \beta = 45^\circ \), then \( (1+\tan \alpha)(1+\tan \beta) = 2 \).

Answer 1

The Correct Option is B

Step 1: Concept Use the sum and product of roots for a quadratic equation and properties of half-angles in a triangle.

Step 2: Meaning Sum = \( \tan(B/2) + \tan(C/2) = -b/a \). Product = \( \tan(B/2) \tan(C/2) = c/a \).

Step 3: Analysis Since \( A = 90^\circ \), \( B + C = 90^\circ \), so \( \frac{B}{2} + \frac{C}{2} = 45^\circ \). \( \tan(\frac{B}{2} + \frac{C}{2}) = \tan(45^\circ) = 1 \). \( \frac{\tan(B/2) + \tan(C/2)}{1 - \tan(B/2)\tan(C/2)} = 1 \implies \frac{-b/a}{1 - c/a} = 1 \). \( -b/a = (a-c)/a \implies -b = a - c \).

Step 4: Conclusion Rearranging gives \( a + b = c \). Final Answer: (B)