The co-ordinates of the vertices of `x^(2) - y^(2) = 1` are

Options

A. `(pm5, 0)`
B. `(pm2, 0)`
C. `(pm1, 0)`
D. `(pm 3, 0)`

Answer

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

### Step 1: Identify the standard form of the hyperbola
The standard form of a hyperbola is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
In our case, the equation is \( x^2 - y^2 = 1 \). We can rewrite it as:
\[ \frac{x^2}{1} - \frac{y^2}{1} = 1 \]
This indicates that \( a^2 = 1 \) and \( b^2 = 1 \).

### Step 2: Determine the values of \( a \) and \( b \)
From \( a^2 = 1 \), we find:
\[ a = \sqrt{1} = 1 \]
Similarly, from \( b^2 = 1 \), we find:
\[ b = \sqrt{1} = 1 \]

### Step 3: Find the coordinates of the vertices
For a hyperbola of the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the coordinates of the vertices are given by:
\[ (\pm a, 0) \]
Substituting the value of \( a \):
\[ (\pm 1, 0) \]
Thus, the coordinates of the vertices are:
\[ (1, 0) \text{ and } (-1, 0) \]

### Final Answer
The coordinates of the vertices of the hyperbola \( x^2 - y^2 = 1 \) are \( (1, 0) \) and \( (-1, 0) \).

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