Final answer:
To find the vertices of the given ellipse, we need to convert the equation into the standard form and complete the squares for both x and y. By analyzing the resulting equation, we can determine the center and major axis. The vertices are located along the major axis at a distance of 5 units from the center.
Step-by-step explanation:
The given equation is -25x2 + 16y2 + 50x + 160y - 25 = 0. To find the vertices, we need to convert the equation into the standard form of an ellipse.
We start by completing the squares for both x and y terms. Subtract 25 from both sides and rearrange the constants to one side:
-25x2 + 50x + 16y2 + 160y = 25.
Divide the coefficients of x and y by -25 and 16 respectively:
x2 - 2x + y2 + 10y = -1.
Now complete the squares for both x and y terms:
(x - 1)2 + (y + 5)2 = -1 + 1 + 25.
Simplify:
(x - 1)2 + (y + 5)2 = 25.
We have an ellipse with the center at (1, -5) and a vertical major axis. The distance from the center to the vertices along the major axis is 5. Therefore, the vertices are (1, -10) and (1, 0).