Find the equation of the auxiliarly circle of the hyperbola `(x^(2))/(6)-(y^(2))/(4) = 1`

Answer

To find the equation of the auxiliary circle of the hyperbola given by

\[ \frac{x^2}{6} - \frac{y^2}{4} = 1, \]

we will follow these steps:

### Step 1: Identify the values of \(a\) and \(b\)

The standard form of a hyperbola is

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1. \]

From the given hyperbola, we can compare and identify:

- \(a^2 = 6\) which gives us \(a = \sqrt{6}\),
- \(b^2 = 4\) which gives us \(b = 2\).

### Step 2: Determine the center of the hyperbola

The center of the hyperbola is at the origin \((0, 0)\).

### Step 3: Find the radius of the auxiliary circle

The radius of the auxiliary circle is equal to \(a\). Since we found \(a = \sqrt{6}\), the radius of the auxiliary circle is \(\sqrt{6}\).

### Step 4: Write the equation of the auxiliary circle

The equation of a circle with center at the origin \((0, 0)\) and radius \(r\) is given by:

\[ x^2 + y^2 = r^2. \]

Substituting \(r = \sqrt{6}\):

\[ x^2 + y^2 = (\sqrt{6})^2 = 6. \]

### Final Answer

Thus, the equation of the auxiliary circle is

\[ x^2 + y^2 = 6. \]

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