Find the equation of Hyperbola with one focus at the origin and directrix `x+3=0` and eccentricity `sqrt3`

Answer

To find the equation of the hyperbola with one focus at the origin (0, 0), a directrix given by the equation \(x + 3 = 0\), and an eccentricity of \(\sqrt{3}\), we can follow these steps:

### Step 1: Identify the components
- Focus \(S\) is at the origin: \(S(0, 0)\)
- Directrix \(D\) is given by \(x + 3 = 0\) or \(x = -3\)
- Eccentricity \(e = \sqrt{3}\)

### Step 2: Define a point on the hyperbola
Let \(P(x, y)\) be a point on the hyperbola. The distance from the focus \(S\) to the point \(P\) is denoted as \(SP\), and the distance from the point \(P\) to the directrix \(D\) is denoted as \(PM\).

### Step 3: Use the definition of hyperbola
According to the definition of a hyperbola, the relationship between these distances is given by:
\[ SP = e \cdot PM \]
Substituting the known values:
\[ SP = \sqrt{3} \cdot PM \]

### Step 4: Calculate \(SP\) and \(PM\)
1. **Calculate \(SP\)**:
\[ SP = \sqrt{(x - 0)^2 + (y - 0)^2} = \sqrt{x^2 + y^2} \]

2. **Calculate \(PM\)**:
The directrix is \(x = -3\). The distance from point \(P(x, y)\) to the directrix is:
\[ PM = |x + 3| \]

### Step 5: Set up the equation
Substituting \(SP\) and \(PM\) into the hyperbola relationship:
\[ \sqrt{x^2 + y^2} = \sqrt{3} \cdot |x + 3| \]

### Step 6: Square both sides
Squaring both sides to eliminate the square root gives:
\[ x^2 + y^2 = 3(x + 3)^2 \]

### Step 7: Expand the right-hand side
Expanding the right-hand side:
\[ x^2 + y^2 = 3(x^2 + 6x + 9) \]
\[ x^2 + y^2 = 3x^2 + 18x + 27 \]

### Step 8: Rearranging the equation
Rearranging the equation to bring all terms to one side:
\[ x^2 + y^2 - 3x^2 - 18x - 27 = 0 \]
\[ -2x^2 + y^2 - 18x - 27 = 0 \]

### Step 9: Multiply through by -1
To make the leading coefficient positive:
\[ 2x^2 - y^2 + 18x + 27 = 0 \]

### Final Equation
Thus, the equation of the hyperbola is:
\[ 2x^2 - y^2 + 18x + 27 = 0 \]