The vertex and focus of a parabola are at a distance of h and k units on positive x-axis from origin. Then equation of parabola is

Options

A. `x^(2) = 4 (k- h) (y-k) `
B. `y^(2) = 4 (k-h) (x-h) `
C. `(x-h)^(2) = 4 (k-h ) (y-k) `
D. `(y- k)^(2) = 4 (k-h) (x-h) `

Answer

To find the equation of a parabola given its vertex and focus, we can follow these steps:

### Step 1: Understand the Position of the Vertex and Focus
Given that the vertex and focus of the parabola are located on the positive x-axis, we can denote the vertex as \( (h, 0) \) and the focus as \( (k, 0) \).

### Step 2: Identify the Distance Between the Vertex and Focus
The distance between the vertex and the focus is given by \( p = k - h \). This distance \( p \) is the distance from the vertex to the focus along the axis of symmetry of the parabola.

### Step 3: Determine the Orientation of the Parabola
Since both the vertex and focus are on the x-axis, the parabola opens horizontally. Specifically, if the focus is to the right of the vertex, the parabola opens to the right.

### Step 4: Write the Standard Form of the Parabola
The standard form of a horizontally opening parabola with vertex at \( (h, 0) \) is given by:
\[ (y - k)^2 = 4p(x - h) \]
where \( p \) is the distance from the vertex to the focus.

### Step 5: Substitute the Values
In our case, since the vertex is at \( (h, 0) \) and \( p = k - h \), we can substitute these values into the equation:
\[ (y - 0)^2 = 4(k - h)(x - h) \]
This simplifies to:
\[ y^2 = 4(k - h)(x - h) \]

### Final Equation
Thus, the equation of the parabola is:
\[ y^2 = 4(k - h)(x - h) \]

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