In ` trianglePQR, angleR = (pi)/(2)`. If ` tan((P)/(2)) " and " tan ((Q)/(2))`are roots of equation, `ax^(2) + bx + c = 0`, then which of the following is true.

Options

A. c = a + b
B. a = b + c
C. b = a + c
D. b = c

Answer

To solve the problem, we will use the properties of the roots of a quadratic equation and some trigonometric identities.

### Step 1: Understand the Problem
We are given a triangle \( PQR \) where \( \angle R = \frac{\pi}{2} \) (90 degrees). The roots of the quadratic equation \( ax^2 + bx + c = 0 \) are \( \tan\left(\frac{P}{2}\right) \) and \( \tan\left(\frac{Q}{2}\right) \).

### Step 2: Use the Sum and Product of Roots
For a quadratic equation \( ax^2 + bx + c = 0 \):
- The sum of the roots \( \tan\left(\frac{P}{2}\right) + \tan\left(\frac{Q}{2}\right) = -\frac{b}{a} \)
- The product of the roots \( \tan\left(\frac{P}{2}\right) \tan\left(\frac{Q}{2}\right) = \frac{c}{a} \)

### Step 3: Apply the Tangent Addition Formula
Using the tangent addition formula:
\[ \tan\left(\frac{P + Q}{2}\right) = \frac{\tan\left(\frac{P}{2}\right) + \tan\left(\frac{Q}{2}\right)}{1 - \tan\left(\frac{P}{2}\right) \tan\left(\frac{Q}{2}\right)} \]
Since \( P + Q + R = \pi \) and \( R = \frac{\pi}{2} \), we have \( P + Q = \frac{\pi}{2} \).

Thus:
\[ \tan\left(\frac{P + Q}{2}\right) = \tan\left(\frac{\pi/2}{2}\right) = \tan\left(\frac{\pi}{4}\right) = 1 \]

### Step 4: Substitute into the Tangent Addition Formula
Substituting into the formula:
\[ 1 = \frac{\tan\left(\frac{P}{2}\right) + \tan\left(\frac{Q}{2}\right)}{1 - \tan\left(\frac{P}{2}\right) \tan\left(\frac{Q}{2}\right)} \]
Cross-multiplying gives:
\[ 1 - \tan\left(\frac{P}{2}\right) \tan\left(\frac{Q}{2}\right) = \tan\left(\frac{P}{2}\right) + \tan\left(\frac{Q}{2}\right) \]

### Step 5: Rearranging the Equation
Rearranging gives:
\[ 1 = \tan\left(\frac{P}{2}\right) + \tan\left(\frac{Q}{2}\right) + \tan\left(\frac{P}{2}\right) \tan\left(\frac{Q}{2}\right) \]
This can be expressed in terms of the coefficients \( a, b, c \):
\[ 1 = -\frac{b}{a} + \frac{c}{a} \]
Multiplying through by \( a \):
\[ a = -b + c \]
Rearranging gives:
\[ a + b = c \]

### Conclusion
Thus, the correct statement is:
\[ a + b = c \]

### Final Answer
The correct option is **option 1: \( a + b = c \)**.