Ellipse

We have,





which is an equation of ellipse.

We have, and and

Equation is an ellipse. , where Eccentricity

For an ellipse, the relationship between the semi-major axis , semi-minor axis , and the focal distance is given by: The given foci are at , so . The eccentricity is defined as: Given that , we solve for : Using : The length of the minor axis is .

We are given the equation of the ellipse: ### Step 1: Group the and Terms Rearrange the equation: ### Step 2: Complete the Square for the -terms For the -terms, , the coefficient of is 6. To complete the square, take half of 6, which is 3, and square it to get 9. Add and subtract 9 inside the parentheses: This becomes: ### Step 3: Complete the Square for the -terms For the -terms, , the coefficient of is -2. To complete the square, take half of -2, which is -1, and square it to get 1. Add and subtract 1 inside the parentheses: ### Step 4: Simplify the Equation Now, simplify the equation: ### Step 5: Identify the Center The equation is now in standard form for an ellipse: From this, we see that the center of the ellipse is at . Thus, the correct answer is:

The length of the latus rectum for an ellipse is given by the formula: where is the semi-major axis and is the semi-minor axis. Here, and , so and . Substituting these values into the formula gives: