The equation of the parabola whose vertex is at the centre of the ellipse `x^2/25+y^2/16`= 1 and the focus coincide with the focus of the ellipse on the positive side of the major axis of the ellipse is

Options

A. `y^2=3x`
B. `y^2=4x`
C. `y^2=5x`
D. `y^2=12x`

Answer

To find the equation of the parabola whose vertex is at the center of the ellipse given by the equation \( \frac{x^2}{25} + \frac{y^2}{16} = 1 \) and whose focus coincides with the focus of the ellipse on the positive side of the major axis, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the center of the ellipse**:
The center of the ellipse given by the equation \( \frac{x^2}{25} + \frac{y^2}{16} = 1 \) is at the point \( (0, 0) \).

**Hint**: The center of an ellipse in standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) is always at the origin \( (0, 0) \).

2. **Determine the semi-major and semi-minor axes**:
From the equation, we can see that \( a^2 = 25 \) and \( b^2 = 16 \). Therefore, \( a = 5 \) and \( b = 4 \).

**Hint**: The values of \( a \) and \( b \) are derived from the denominators of the ellipse equation.

3. **Find the foci of the ellipse**:
The foci of the ellipse are located at \( (c, 0) \) and \( (-c, 0) \), where \( c = \sqrt{a^2 - b^2} \).
\[ c = \sqrt{25 - 16} = \sqrt{9} = 3 \]
Thus, the foci are at \( (3, 0) \) and \( (-3, 0) \).

**Hint**: The distance to the foci \( c \) can be found using the formula \( c = \sqrt{a^2 - b^2} \).

4. **Identify the focus of the parabola**:
Since the focus of the parabola coincides with the focus of the ellipse on the positive side of the major axis, the focus of the parabola is at \( (3, 0) \).

**Hint**: The focus of the parabola is crucial for determining its equation.

5. **Write the equation of the parabola**:
The standard form of a parabola that opens to the right is given by:
\[ y^2 = 4ax \]
Here, the vertex is at the origin \( (0, 0) \) and the focus is at \( (a, 0) = (3, 0) \). Thus, \( a = 3 \).

**Hint**: The parameter \( a \) in the equation \( y^2 = 4ax \) represents the distance from the vertex to the focus.

6. **Substituting the value of \( a \)**:
Substitute \( a = 3 \) into the equation:
\[ y^2 = 4 \cdot 3 \cdot x \]
This simplifies to:
\[ y^2 = 12x \]

**Hint**: Make sure to simplify the equation correctly after substituting the value of \( a \).

### Final Answer:
The equation of the parabola is \( y^2 = 12x \).