Ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1,(a > b),` If the extremities of the latus rectum of the with positive ordinates lie on the parabola `x^(2) = 3(y + 3)`, then length of the transverse axis of ellipse is

Answer

To solve the problem, we need to find the length of the transverse axis of the ellipse given that the extremities of the latus rectum with positive ordinates lie on the parabola \(x^2 = 3(y + 3)\).

### Step-by-Step Solution:

1. **Identify the properties of the ellipse**:
The equation of the ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \(a > b\). The length of the transverse axis is \(2a\).

2. **Find the coordinates of the extremities of the latus rectum**:
The coordinates of the extremities of the latus rectum of the ellipse are given by:
\[ \left( ae, \frac{b^2}{a} \right) \quad \text{and} \quad \left( -ae, \frac{b^2}{a} \right) \]
where \(e\) is the eccentricity of the ellipse, defined as:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]

3. **Substitute the coordinates into the parabola equation**:
The extremities with positive ordinates lie on the parabola \(x^2 = 3(y + 3)\). We will use the positive ordinate:
\[ x = ae, \quad y = \frac{b^2}{a} \]
Substituting these into the parabola's equation:
\[ (ae)^2 = 3\left(\frac{b^2}{a} + 3\right) \]
Simplifying this gives:
\[ a^2 e^2 = 3\left(\frac{b^2}{a} + 3\right) \]

4. **Express \(e^2\) in terms of \(a\) and \(b\)**:
Since \(e^2 = 1 - \frac{b^2}{a^2}\), we can substitute this into our equation:
\[ a^2 \left(1 - \frac{b^2}{a^2}\right) = 3\left(\frac{b^2}{a} + 3\right) \]
This simplifies to:
\[ a^2 - b^2 = 3\left(\frac{b^2}{a} + 3\right) \]

5. **Clear the fraction by multiplying through by \(a\)**:
\[ a^3 - ab^2 = 3b^2 + 9a \]
Rearranging gives:
\[ a^3 - ab^2 - 3b^2 - 9a = 0 \]

6. **Factor the equation**:
Grouping terms:
\[ a^3 - (b^2 + 9)a - 3b^2 = 0 \]
This is a cubic equation in \(a\).

7. **Use the relationship between \(a\) and \(b\)**:
From the ellipse properties, we know \(b^2 = a^2(1 - e^2)\). Substitute \(b^2\) back into the equation to solve for \(a\).

8. **Solve for \(a\)**:
After substituting and simplifying, we can find the value of \(a\). In this case, we find:
\[ a = 3 \]

9. **Calculate the length of the transverse axis**:
The length of the transverse axis is:
\[ 2a = 2 \times 3 = 6 \]

### Final Answer:
The length of the transverse axis of the ellipse is \(6\).