If a `sectheta = 1 -b tan theta and a^2 sec^2 theta = 5 + b^2 tan^2 theta,` then (i) `a^2b^2-4a^2=9b^2` (ii) `a^2b^2+4a^2=9b^2` (iii) `a^2b^2+9b^2=4a^2` (iv) `a^2b^2+9b^2=5a^2`

Options

A. `a ^(2) b ^(2) - 4a ^(2) = 9b ^(2)`
B. `a^(2) b^(2) + 4a ^(2)= 9b ^(2)`
C. `a ^(2)b ^(2) + 9b ^(2)= 4a^(2)`
D. `4a ^(2) b ^(2) + 9b ^(2) = 5a ^(2)`

Answer

To solve the given problem, we start with the equations provided:

1. \( a \sec \theta = 1 - b \tan \theta \)
2. \( a^2 \sec^2 \theta = 5 + b^2 \tan^2 \theta \)

### Step 1: Rewrite the first equation
From the first equation, we can express \( \sec \theta \) in terms of \( \tan \theta \):
\[ \sec \theta = \frac{1 - b \tan \theta}{a} \]

### Step 2: Square both sides of the first equation
Now, we square both sides to find \( \sec^2 \theta \):
\[ \sec^2 \theta = \left(\frac{1 - b \tan \theta}{a}\right)^2 = \frac{(1 - b \tan \theta)^2}{a^2} \]

Expanding the right side:
\[ \sec^2 \theta = \frac{1 - 2b \tan \theta + b^2 \tan^2 \theta}{a^2} \]

### Step 3: Substitute into the second equation
Now, substitute \( \sec^2 \theta \) into the second equation:
\[ a^2 \left(\frac{1 - 2b \tan \theta + b^2 \tan^2 \theta}{a^2}\right) = 5 + b^2 \tan^2 \theta \]

This simplifies to:
\[ 1 - 2b \tan \theta + b^2 \tan^2 \theta = 5 + b^2 \tan^2 \theta \]

### Step 4: Rearrange the equation
Subtract \( b^2 \tan^2 \theta \) from both sides:
\[ 1 - 2b \tan \theta = 5 \]

### Step 5: Solve for \( \tan \theta \)
Rearranging gives:
\[ -2b \tan \theta = 4 \]
\[ \tan \theta = -\frac{2}{b} \]

### Step 6: Substitute \( \tan \theta \) back into \( \sec \theta \)
Now substitute \( \tan \theta \) back into the expression for \( \sec \theta \):
\[ \sec \theta = \frac{1 - b \left(-\frac{2}{b}\right)}{a} = \frac{1 + 2}{a} = \frac{3}{a} \]

### Step 7: Use the identity \( \sec^2 \theta - \tan^2 \theta = 1 \)
Using the identity \( \sec^2 \theta - \tan^2 \theta = 1 \):
\[ \left(\frac{3}{a}\right)^2 - \left(-\frac{2}{b}\right)^2 = 1 \]
\[ \frac{9}{a^2} - \frac{4}{b^2} = 1 \]

### Step 8: Rearranging the equation
Multiply through by \( a^2 b^2 \) to eliminate the denominators:
\[ 9b^2 - 4a^2 = a^2 b^2 \]

### Step 9: Rearranging to find the final equation
Rearranging gives:
\[ a^2 b^2 - 4a^2 = 9b^2 \]

### Conclusion
Thus, we have derived the equation:
\[ a^2 b^2 - 4a^2 = 9b^2 \]

This corresponds to option (i).