The equation `2x^2 - 3y^2 -6 = 0`represents a:

Options

A. Circle
B. parabola
C. ellipse
D. hyperbola

Answer

To determine what the equation \(2x^2 - 3y^2 - 6 = 0\) represents, we can follow these steps:

### Step 1: Rearrange the equation
First, we start with the given equation:
\[ 2x^2 - 3y^2 - 6 = 0 \]
We can rearrange it to isolate the terms involving \(x\) and \(y\):
\[ 2x^2 - 3y^2 = 6 \]

### Step 2: Divide by 6
Next, we divide the entire equation by 6 to simplify it:
\[ \frac{2x^2}{6} - \frac{3y^2}{6} = 1 \]
This simplifies to:
\[ \frac{x^2}{3} - \frac{y^2}{2} = 1 \]

### Step 3: Identify the form of the equation
Now, we can see that the equation is in the standard form of a hyperbola:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
where \(a^2 = 3\) and \(b^2 = 2\). This confirms that the equation represents a hyperbola that opens along the x-axis.

### Conclusion
Thus, the equation \(2x^2 - 3y^2 - 6 = 0\) represents a hyperbola.

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