If `x cos theta + y sin theta =2` and `x cos theta - y sin theta =0`, then which one of the following is true ?

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 problem, we have two equations:

1. \( x \cos \theta + y \sin \theta = 2 \) (Equation 1)
2. \( x \cos \theta - y \sin \theta = 0 \) (Equation 2)

We need to find the values of \( x \) and \( y \) and then determine which of the given options is true.

### Step 1: Add the two equations
Let's add Equation 1 and Equation 2:

\[ (x \cos \theta + y \sin \theta) + (x \cos \theta - y \sin \theta) = 2 + 0 \]

This simplifies to:

\[ 2x \cos \theta = 2 \]

### Step 2: Solve for \( x \)
Now, divide both sides by \( 2 \):

\[ x \cos \theta = 1 \]

Now, divide both sides by \( \cos \theta \):

\[ x = \frac{1}{\cos \theta} \]

### Step 3: Substitute \( x \) back into Equation 2
Now, we will substitute the value of \( x \) into Equation 2 to find \( y \):

\[ \frac{1}{\cos \theta} \cos \theta - y \sin \theta = 0 \]

This simplifies to:

\[ 1 - y \sin \theta = 0 \]

### Step 4: Solve for \( y \)
Now, rearranging gives us:

\[ y \sin \theta = 1 \]

Now, divide both sides by \( \sin \theta \):

\[ y = \frac{1}{\sin \theta} \]

### Step 5: Find \( x^2 + y^2 \)
Now we have:

\[ x = \frac{1}{\cos \theta}, \quad y = \frac{1}{\sin \theta} \]

Now we can find \( x^2 + y^2 \):

\[ x^2 = \left(\frac{1}{\cos \theta}\right)^2 = \frac{1}{\cos^2 \theta} \]

\[ y^2 = \left(\frac{1}{\sin \theta}\right)^2 = \frac{1}{\sin^2 \theta} \]

Thus,

\[ x^2 + y^2 = \frac{1}{\cos^2 \theta} + \frac{1}{\sin^2 \theta} \]

### Step 6: Combine the fractions
To combine these fractions, we find a common denominator:

\[ x^2 + y^2 = \frac{\sin^2 \theta + \cos^2 \theta}{\sin^2 \theta \cos^2 \theta} \]

Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \):

\[ x^2 + y^2 = \frac{1}{\sin^2 \theta \cos^2 \theta} \]

### Step 7: Find \( \frac{1}{x^2} + \frac{1}{y^2} \)
Now let's find \( \frac{1}{x^2} + \frac{1}{y^2} \):

\[ \frac{1}{x^2} = \cos^2 \theta \]

\[ \frac{1}{y^2} = \sin^2 \theta \]

Thus,

\[ \frac{1}{x^2} + \frac{1}{y^2} = \cos^2 \theta + \sin^2 \theta = 1 \]

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