The focus at (10, 0) the directrix `x= -10`.

Answer

To find the equation of the parabola with the given focus at (10, 0) and the directrix x = -10, we can follow these steps:

### Step 1: Identify the Focus and Directrix
- The focus of the parabola is given as \( F(10, 0) \).
- The directrix is given as \( x = -10 \).

### Step 2: Find the Vertex
- The vertex of the parabola lies midway between the focus and the directrix.
- The x-coordinate of the vertex can be calculated as the average of the x-coordinates of the focus and the directrix:
\[ x_{vertex} = \frac{x_{focus} + x_{directrix}}{2} = \frac{10 + (-10)}{2} = \frac{0}{2} = 0 \]
- The y-coordinate of the vertex is the same as that of the focus since both points lie on the same horizontal line (y = 0):
\[ y_{vertex} = 0 \]
- Therefore, the vertex \( V \) is at \( (0, 0) \).

### Step 3: Determine the Orientation of the Parabola
- Since the focus is to the right of the directrix, the parabola opens to the right.

### Step 4: Find the Distance \( a \)
- The distance \( a \) is the distance from the vertex to the focus:
\[ a = x_{focus} - x_{vertex} = 10 - 0 = 10 \]

### Step 5: Write the Equation of the Parabola
- The standard form of the equation of a parabola that opens to the right is:
\[ y^2 = 4ax \]
- Substituting \( a = 10 \):
\[ y^2 = 4 \cdot 10 \cdot x = 40x \]

### Final Equation
The equation of the parabola is:
\[ y^2 = 40x \]

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