Given ` x cos theta + y sin theta = 2 and x cos theta - y sin theta = 0 ` , then which of the following is correct

Options

A. ` x^(2) + y^(2) = 1 `
B. ` (1)/( x^(2)) + (1)/(y^(2)) = 1 `
C. xy = 1
D. ` x^(2) - y^(2) = 1 `

Answer

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

### Step 1: Write down the equations
We have two equations:
1. \( x \cos \theta + y \sin \theta = 2 \) (Equation 1)
2. \( x \cos \theta - y \sin \theta = 0 \) (Equation 2)

### Step 2: Solve Equation 2 for \( x \cos \theta \)
From Equation 2, we can express \( x \cos \theta \) in terms of \( y \sin \theta \):
\[ x \cos \theta = y \sin \theta \]

### Step 3: Substitute \( x \cos \theta \) in Equation 1
Now, substitute \( x \cos \theta \) from Equation 2 into Equation 1:
\[ y \sin \theta + y \sin \theta = 2 \]
This simplifies to:
\[ 2y \sin \theta = 2 \]

### Step 4: Solve for \( y \)
Dividing both sides by 2:
\[ y \sin \theta = 1 \]
Thus, we can find \( y \):
\[ y = \frac{1}{\sin \theta} \]

### Step 5: Substitute \( y \) back to find \( x \)
Now, substitute \( y \) back into Equation 2 to find \( x \):
\[ x \cos \theta = y \sin \theta = \frac{1}{\sin \theta} \sin \theta = 1 \]
Thus, we can find \( x \):
\[ x = \frac{1}{\cos \theta} \]

### Step 6: Check the options
Now we have:
- \( x = \frac{1}{\cos \theta} \)
- \( y = \frac{1}{\sin \theta} \)

We will check the options provided in the question.

#### Option 1: \( x^2 + y^2 = 1 \)
Calculating:
\[ x^2 + y^2 = \left(\frac{1}{\cos \theta}\right)^2 + \left(\frac{1}{\sin \theta}\right)^2 = \frac{1}{\cos^2 \theta} + \frac{1}{\sin^2 \theta} \]
This does not equal 1, so this option is incorrect.

#### Option 2: \( \frac{1}{x^2} + \frac{1}{y^2} = 1 \)
Calculating:
\[ \frac{1}{x^2} + \frac{1}{y^2} = \cos^2 \theta + \sin^2 \theta = 1 \]
This option is correct.

#### Option 3: \( xy = 1 \)
Calculating:
\[ xy = \left(\frac{1}{\cos \theta}\right) \left(\frac{1}{\sin \theta}\right) = \frac{1}{\cos \theta \sin \theta} \]
This does not equal 1, so this option is incorrect.

#### Option 4: \( x + y = 1 \)
Calculating:
\[ x + y = \frac{1}{\cos \theta} + \frac{1}{\sin \theta} \]
This does not equal 1, so this option is incorrect.

### Conclusion
The correct option is **Option 2: \( \frac{1}{x^2} + \frac{1}{y^2} = 1 \)**.