Question

An ellipse is graphed. Which statements about the ellipse are true? Select three options.

Answer 1

Answer:

The true statements are;

1) The center of the ellipse is at (-2, -5)

3) The covertices are at (-2, -4) and (-2, -6)

4) The distance between the center and each focus is 2·√6 units

Step-by-step explanation:

The given graph of the ellipse gives;

1) The location of the center of the ellipse = (-2, -5)

2) The location of the vertices of the ellipse = (-7, -5) and (3, -5)

The distance between the center (-2, -5) and each vertex, d, is given as follows;

d = -2 - (-7) = 5 and d = 3 - (-2) = 5

Therefore, the distance between the center (-2, -5) and each vertex is 5 units

3) The location of the covertices, given in the diagram are (-2, -4), and (-2, -6)

4) The coordinates of the focus of an ellipse = h - c, k

c = √(a² + b²)

Where, for the given ellipse, h = -2, k = -5, c = √(5² - 1²) = √(24) = 2·√6

∴ The location of the focus of the ellipse = (-2 - 2·√6, -5)

The distance between the center and each focus, d = -2 - (-2 - 2·√6) = 2·√6

5) The x-coordinate of the directrices of the ellipse = ± a/e

Where; a = 5

b² = a²·(1 - e²)

∴ e² = 1 - b²/a² = 1 - 1/25 = 24/25

e = 2·√6/5

The directrices = -2 ± 5/(2·√6/5) ≈ -2 ± 5.1

The distance between the center and the directrices ≈ 5.1 units

Therefore, the directrices are vertical lines approximately 5.1 units from the center