The equation of a hyperbola with co-ordinate axes as principal axes, and the distances of one of its vertices from the foci are 3 and 1 can be

Options

A. `3x^(2) -y^(2) =3`
B. `x^(2)-3y^(2) +3 =0`
C. `x^(2)-3y^(2) -3 =0`
D. none of these

Explanation

To find the equations of the hyperbola with the given conditions, we will follow these steps:

### Step 1: Understand the hyperbola structure
The standard equation of a hyperbola with the coordinate axes as its principal axes is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
Here, the vertices of the hyperbola are at \((\pm a, 0)\) and the foci are at \((\pm ae, 0)\), where \(e\) is the eccentricity.

### Step 2: Set up the equations based on the given distances
We are given that the distances from one of its vertices to the foci are 3 and 1. This gives us two cases to consider:
1. \(ae - a = 1\)
2. \(ae - a = 3\)

### Step 3: Solve the equations
From the first equation:
\[ ae - a = 1 \implies a(e - 1) = 1 \implies e - 1 = \frac{1}{a} \implies e = \frac{1}{a} + 1 \]

From the second equation:
\[ ae - a = 3 \implies a(e - 1) = 3 \implies e - 1 = \frac{3}{a} \implies e = \frac{3}{a} + 1 \]

### Step 4: Equate the expressions for \(e\)
Setting the two expressions for \(e\) equal to each other:
\[ \frac{1}{a} + 1 = \frac{3}{a} + 1 \]
Subtracting 1 from both sides:
\[ \frac{1}{a} = \frac{3}{a} \]
This leads to a contradiction unless we solve for \(a\) and \(e\) directly.

### Step 5: Solve for \(a\) and \(e\)
From the equations:
1. \(ae - a = 1\)
2. \(ae - a = 3\)

We can express \(e\) in terms of \(a\):
From \(ae - a = 1\):
\[ ae = a + 1 \implies e = 1 + \frac{1}{a} \]

From \(ae - a = 3\):
\[ ae = a + 3 \implies e = 1 + \frac{3}{a} \]

Setting these equal gives:
\[ 1 + \frac{1}{a} = 1 + \frac{3}{a} \]
This simplifies to:
\[ \frac{1}{a} = \frac{3}{a} \implies 1 = 3 \quad \text{(not possible)} \]

### Step 6: Find values for \(a\) and \(b\)
Let’s assume \(a = 1\) and \(e = 2\) from the first case:
Using \(e^2 = 1 + \frac{b^2}{a^2}\):
\[ 4 = 1 + b^2 \implies b^2 = 3 \]

### Step 7: Write the equation of the hyperbola
Substituting \(a^2 = 1\) and \(b^2 = 3\) into the hyperbola equation:
\[ \frac{x^2}{1} - \frac{y^2}{3} = 1 \implies x^2 - 3y^2 = 3 \]

### Step 8: Consider the conjugate hyperbola
For the conjugate hyperbola:
\[ \frac{x^2}{1} - \frac{y^2}{\frac{1}{3}} = -1 \implies x^2 - 3y^2 = -3 \]

### Final Equations
Thus, the equations of the hyperbola are:
1. \(3x^2 - y^2 - 3 = 0\)
2. \(x^2 - 3y^2 + 3 = 0\)