Find the vertices, foci, and directrices of the given ellipse.
A) vertices ; foci (0, -3), (-14, -3); directrices
B) vertices ; foci (0, -3), (-14, -3); directrices
C) vertices (0, -3), (-14, -3); foci ; directrices
D) vertices (0, -3), (-14, -3); foci ; directrices

Question

Find the vertices, foci, and directrices of the given ellipse. 
A) vertices ; foci (0, -3), (-14, -3); directrices 
B) vertices ; foci (0, -3), (-14, -3); directrices 
C) vertices (0, -3), (-14, -3); foci ; directrices 
D) vertices (0, -3), (-14, -3); foci ; directrices

Quizplus · Accepted Answer

The vertices are the endpoints of the major axis of the ellipse, which is the line segment connecting (-14, -3) and (0, -3), so the vertices are (0, -3) and (-14, -3). The foci are the points on the major axis that are a distance of 5 units away from the center of the ellipse, which is halfway between the vertices at (-7, -3). Since the major axis has length 14, the distance between the foci is 2*sqrt(5^2 - 7^2) = 6. Therefore, the foci are (-7-3, -3) = (-10, -3) and (-7+3, -3) = (-4, -3). The directrices are the lines perpendicular to the major axis that pass through the foci. Since the major axis is horizontal (parallel to the x-axis), the directrices are the vertical lines x = -10 and x = -4. Thus, the answer is C.

Find the Vertices, Foci, and Directrices of the Given Ellipse