The equation of the parabola whose vertex and focus lie on the axis of xat distances aand `a_1` from the origin respectively, is

Options

A. `y^2 = 4(a_1 -a) x`
B. `y^2=4(a_1-a)(x-a)`
C. `y^2=4(a_1-a)(x-a_1)`
D. none of these

Answer

To find the equation of the parabola whose vertex and focus lie on the x-axis at distances \( a \) and \( a_1 \) from the origin respectively, we can follow these steps:

### Step 1: Identify the Coordinates of the Vertex and Focus
The vertex \( V \) of the parabola is located at \( (a, 0) \) and the focus \( F \) is at \( (a_1, 0) \).

### Step 2: Determine the Distance Between the Vertex and the Focus
The distance between the vertex and the focus is given by:
\[ CF = a_1 - a \]
where \( CF \) is the distance from the vertex to the focus.

### Step 3: Write the Standard Form of the Parabola
The standard form of the equation of a parabola that opens to the right (with vertex at \( (h, k) \)) is:
\[ (y - k)^2 = 4p(x - h) \]
where \( p \) is the distance from the vertex to the focus.

### Step 4: Substitute the Values into the Standard Form
In our case:
- The vertex \( (h, k) = (a, 0) \)
- The distance \( p = a_1 - a \)

Substituting these values into the standard form gives:
\[ (y - 0)^2 = 4(a_1 - a)(x - a) \]

### Step 5: Simplify the Equation
This simplifies to:
\[ y^2 = 4(a_1 - a)(x - a) \]

### Final Result
Thus, the equation of the parabola is:
\[ y^2 = 4(a_1 - a)(x - a) \]

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