The equation to the pair of asymptotes of the hyperola `2x^(2)-y^(2)=1` is

Options

A. `2x^(2)+y^(2)=0`
B. `2x^(2)-y^(2)=0`
C. `x^(2)+2y^(2)=0`
D. `x^(2)-2y^(2)=0`

Answer

To find the equations of the pair of asymptotes of the hyperbola given by the equation \(2x^2 - y^2 = 1\), we can follow these steps:

### Step 1: Rewrite the Hyperbola Equation
We start with the hyperbola equation:
\[ 2x^2 - y^2 = 1 \]
We can rearrange this equation to bring 1 to the left side:
\[ 2x^2 - y^2 - 1 = 0 \]

### Step 2: Understand the Concept of Asymptotes
The asymptotes of a hyperbola can be found by replacing the constant term (which is 1 in this case) with a variable, say \(\lambda\). So, we replace the constant in the hyperbola equation:
\[ 2x^2 - y^2 + \lambda = 0 \]

### Step 3: Use the Asymptote Condition
To find the asymptotes, we need to use the condition that the equation must hold true for \(\lambda = 0\). We will use the general condition for the asymptotes derived from the conic section equation:
\[ ABC + 2FGH - AF^2 - BG^2 - CH^2 = 0 \]
where \(A\), \(B\), \(C\), \(F\), \(G\), and \(H\) are coefficients from the equation of the conic section.

### Step 4: Identify Coefficients
From our equation \(2x^2 - y^2 + \lambda = 0\), we can identify:
- \(A = 2\)
- \(B = -1\)
- \(C = \lambda\)
- \(F = 0\) (no \(xy\) term)
- \(G = 0\) (no \(x\) term)
- \(H = 0\) (no \(y\) term)

### Step 5: Substitute Coefficients into the Condition
Now we substitute these values into the condition:
\[ 2 \cdot (-1) \cdot \lambda + 2 \cdot 0 \cdot 0 - 2 \cdot 0^2 - (-1) \cdot 0^2 - \lambda \cdot 0^2 = 0 \]
This simplifies to:
\[ -2\lambda = 0 \]

### Step 6: Solve for \(\lambda\)
From the equation \(-2\lambda = 0\), we find:
\[ \lambda = 0 \]

### Step 7: Write the Asymptote Equations
Now, we substitute \(\lambda = 0\) back into the equation for the asymptotes:
\[ 2x^2 - y^2 + 0 = 0 \]
This simplifies to:
\[ 2x^2 - y^2 = 0 \]

### Conclusion
Thus, the equations of the pair of asymptotes of the hyperbola \(2x^2 - y^2 = 1\) are:
\[ 2x^2 - y^2 = 0 \]