Find the equation of the parabola having the vertex at the origin, the directrix is parallel to y - axis and passes through (-3,6).

Answer

To find the equation of the parabola with the given conditions, we can follow these steps:

### Step 1: Identify the standard form of the parabola
Since the vertex is at the origin (0, 0) and the directrix is parallel to the y-axis, the standard form of the equation of the parabola can be expressed as:
\[ y^2 = 4ax \]
where \(a\) is the distance from the vertex to the focus.

### Step 2: Use the point that the parabola passes through
The parabola passes through the point (-3, 6). We will substitute \(x = -3\) and \(y = 6\) into the equation:
\[ 6^2 = 4a(-3) \]

### Step 3: Simplify the equation
Calculating \(6^2\):
\[ 36 = 4a(-3) \]
This simplifies to:
\[ 36 = -12a \]

### Step 4: Solve for \(a\)
To find \(a\), divide both sides by -12:
\[ a = \frac{36}{-12} = -3 \]

### Step 5: Substitute \(a\) back into the standard form
Now that we have \(a\), we can substitute it back into the standard form of the parabola:
\[ y^2 = 4(-3)x \]
This simplifies to:
\[ y^2 = -12x \]

### Final Equation
Thus, the equation of the parabola is:
\[ y^2 = -12x \]