Equation of parabola with vertex (0,0) and focus (2,0) is

Options

A. `x^(2) = x `
B. `y^(2) = 2x `
C. `y^(2) = 4x `
D. `y^(2) = 8x `

Answer

To find the equation of a parabola with a vertex at (0,0) and a focus at (2,0), we can follow these steps:

### Step 1: Understand the properties of the parabola
A parabola is defined as the set of all points that are equidistant from a point called the focus and a line called the directrix. The vertex is the midpoint between the focus and the directrix.

### Step 2: Identify the orientation of the parabola
Since the focus (2,0) is to the right of the vertex (0,0), the parabola opens to the right.

### Step 3: Use the standard form of the parabola
The standard equation of a parabola that opens to the right with vertex at (0,0) is given by:
\[ y^2 = 4ax \]
where \(a\) is the distance from the vertex to the focus.

### Step 4: Determine the value of \(a\)
In this case, the distance from the vertex (0,0) to the focus (2,0) is 2 units. Therefore, we have:
\[ a = 2 \]

### Step 5: Substitute \(a\) into the equation
Now, substituting \(a\) into the standard equation:
\[ y^2 = 4 \cdot 2 \cdot x \]
This simplifies to:
\[ y^2 = 8x \]

### Step 6: Write the final equation
Thus, the equation of the parabola with vertex (0,0) and focus (2,0) is:
\[ \boxed{y^2 = 8x} \]

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