Question

The center of an ellipse is (1,6). One focus of the ellipse is (-2,6). One vertex of the ellipse is (10,6). What is the equation of the ellipse in standard form?

1) x²/25 + y²/9 = 1
2) x²/9 + y²/25 = 1
3) x²/16 + y²/4 = 1
4) x²/4 + y²/16 = 1

Answer 1

The equation of the ellipse in standard form is x²/4 + y²/16 = 1.

Step-by-step explanation:

To determine the equation of the ellipse in standard form, we can use the distance formula. The distance between the center of the ellipse and one of its foci is called the distance from the center to the focus (c). The distance between the center of the ellipse and one of its vertices is called the distance from the center to the vertex (a). The equation of an ellipse with center (h, k), horizontal major axis, and length 2a, and vertical minor axis and length 2b is given by:

((x-h)^2)/(a^2) + ((y-k)^2)/(b^2) = 1

From the given information, we can determine that the center is (1, 6), the focus is (-2, 6), and the vertex is (10, 6). Since the x-coordinates of the center, focus, and vertex are given, we can determine that the length of the major axis is 2a = 9. Therefore, a = 4.5. Since the y-coordinates of the center, focus, and vertex are the same, we can determine that the length of the minor axis is 2b = 0, which means b = 0.

Substituting the values into the equation, we get:

((x-1)^2)/(4.5^2) + ((y-6)^2)/(0^2) = 1

Simplifying further, we get:

((x-1)^2)/(20.25) + ((y-6)^2)/(0) = 1

Since we cannot divide by zero, the equation becomes:

((x-1)^2)/(20.25) = 1

Therefore, the equation of the ellipse in standard form is:

x²/20.25 + y²/0 = 1

4) x²/4 + y²/16 = 1