The equation of parabola through (-1,3) and symmetric with respect to x-axis and vertex at origin is (i) `y^(2) = - 9x` (ii) ` y^(2) = 9 x ` (iii) ` y^(2) = 3 x` (iv) ` y^(2) = - 3x`

Options

A. `y^(2) = - 9x`
B. ` y^(2) = 9 x `
C. ` y^(2) = 3 x`
D. ` y^(2) = - 3x`

Answer

To find the equation of the parabola that passes through the point (-1, 3), is symmetric with respect to the x-axis, and has its vertex at the origin, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Standard Form of the Parabola**:
Since the parabola is symmetric with respect to the x-axis and has its vertex at the origin, its equation can be expressed in the form:
\[ y^2 = 4ax \]
or
\[ y^2 = -4ax \]
depending on whether it opens to the right or left.

2. **Use the Given Point**:
The parabola passes through the point (-1, 3). We can substitute \( x = -1 \) and \( y = 3 \) into the equation to find the value of \( a \).

3. **Substituting the Point into the Equation**:
Let's use the equation \( y^2 = -4ax \) since we are looking for a parabola that opens to the left (as it is symmetric with respect to the x-axis):
\[ 3^2 = -4a(-1) \]
This simplifies to:
\[ 9 = 4a \]

4. **Solve for \( a \)**:
Rearranging the equation gives:
\[ a = \frac{9}{4} \]

5. **Write the Equation of the Parabola**:
Now, substituting the value of \( a \) back into the standard form:
\[ y^2 = -4\left(\frac{9}{4}\right)x \]
This simplifies to:
\[ y^2 = -9x \]

6. **Conclusion**:
Therefore, the equation of the parabola is:
\[ y^2 = -9x \]
The correct option is (i) \( y^2 = -9x \).