The equation of the parabola whose focus is at the origine is `y^2=4a(x+a)`.

Answer

To derive the equation of the parabola whose focus is at the origin, we start with the given information and follow these steps:

### Step 1: Understanding the Parabola
The standard form of a parabola that opens to the right is given by the equation:
\[ y^2 = 4ax \]
where \(a\) is the distance from the vertex to the focus.

### Step 2: Identify the Focus and Directrix
In this case, we know that the focus is at the origin (0, 0). The directrix of the parabola is a vertical line located at \(x = -a\).

### Step 3: Determine the Vertex
The vertex of the parabola is the midpoint between the focus and the directrix. Therefore, the vertex is located at:
\[ \left(-\frac{a}{2}, 0\right) \]

### Step 4: Equation of the Directrix
The equation of the directrix, which is a vertical line, is given by:
\[ x = -a \]

### Step 5: Use the Definition of a Parabola
By definition, a parabola is the set of all points \(P(x, y)\) such that the distance from \(P\) to the focus is equal to the distance from \(P\) to the directrix.

1. **Distance from \(P\) to the focus (0, 0)**:
\[ \text{Distance} = \sqrt{x^2 + y^2} \]

2. **Distance from \(P\) to the directrix \(x = -a\)**:
\[ \text{Distance} = |x + a| \]

### Step 6: Set the Distances Equal
Setting the two distances equal gives us:
\[ \sqrt{x^2 + y^2} = |x + a| \]

### Step 7: Square Both Sides
To eliminate the square root, we square both sides:
\[ x^2 + y^2 = (x + a)^2 \]

### Step 8: Expand the Right Side
Expanding the right side:
\[ x^2 + y^2 = x^2 + 2ax + a^2 \]

### Step 9: Simplify the Equation
Subtract \(x^2\) from both sides:
\[ y^2 = 2ax + a^2 \]

### Step 10: Rearranging the Equation
Rearranging gives us:
\[ y^2 = 4a\left(\frac{1}{4}x + \frac{a}{4}\right) \]
which can be rewritten as:
\[ y^2 = 4a\left(x + a\right) \]

### Conclusion
Thus, we have derived the equation of the parabola whose focus is at the origin:
\[ y^2 = 4a(x + a) \]