What are the foci of the ellipse given by the equation 25x^2 + 16y^2= 400?
To find the foci of the ellipse, we first need to rewrite the equation in standard form.
Dividing both sides of the equation by 400, we get:
x^2/16 + y^2/25 = 1
Now we can see that this ellipse is centered at the origin (0,0), and because the denominator of x^2 is smaller than the denominator of y^2, we know that the major axis is along the y-axis.
The equation for the distance from the center to the foci is given by c = sqrt(a^2 - b^2), where a = 25 (the square root of the denominator of y^2) and b = 16 (the square root of the denominator of x^2).
Plugging in the values, we get:
c = sqrt(25^2 - 16^2) = sqrt(625 - 256) = sqrt(369) = 19
So the distance from the center to each focus is 19 units. Since the center is at the origin, the foci are therefore at (0,19) and (0,-19).
Therefore, the foci of the ellipse given by the equation 25x^2 + 16y^2 = 400 are at (0,19) and (0,-19).