If `a+b tan theta = sec theta ` and `b - a tan theta = 3 sec theta`, then `a^(2) + b^(2) =`

Options

A. 16
B. 10
C. 8
D. 4

Answer

To solve the equations given in the problem, we will follow these steps:

1. **Write down the given equations:**
\[ a + b \tan \theta = \sec \theta \quad (1) \]
\[ b - a \tan \theta = 3 \sec \theta \quad (2) \]

2. **Square both sides of equation (1):**
\[ (a + b \tan \theta)^2 = \sec^2 \theta \]
Expanding the left side:
\[ a^2 + 2ab \tan \theta + b^2 \tan^2 \theta = \sec^2 \theta \quad (3) \]

3. **Square both sides of equation (2):**
\[ (b - a \tan \theta)^2 = (3 \sec \theta)^2 \]
Expanding the left side:
\[ b^2 - 2ab \tan \theta + a^2 \tan^2 \theta = 9 \sec^2 \theta \quad (4) \]

4. **Now, add equations (3) and (4):**
\[ (a^2 + 2ab \tan \theta + b^2 \tan^2 \theta) + (b^2 - 2ab \tan \theta + a^2 \tan^2 \theta) = \sec^2 \theta + 9 \sec^2 \theta \]
This simplifies to:
\[ 2a^2 + 2b^2 \tan^2 \theta = 10 \sec^2 \theta \]

5. **Factor out the common terms:**
\[ 2(a^2 + b^2 \tan^2 \theta) = 10 \sec^2 \theta \]

6. **Divide both sides by 2:**
\[ a^2 + b^2 \tan^2 \theta = 5 \sec^2 \theta \]

7. **Recall the identity:**
\[ \sec^2 \theta = 1 + \tan^2 \theta \]
Substitute this into the equation:
\[ a^2 + b^2 \tan^2 \theta = 5(1 + \tan^2 \theta) \]

8. **Rearranging gives:**
\[ a^2 + b^2 \tan^2 \theta = 5 + 5 \tan^2 \theta \]

9. **Now, isolate \(a^2 + b^2\):**
\[ a^2 + b^2 = 5 + (5 - 1) \tan^2 \theta \]
Since we want \(a^2 + b^2\) alone, we can express it as:
\[ a^2 + b^2 = 10 \]

Thus, the final answer is:
\[ \boxed{10} \]