Find the foci of the ellipse with the equation (16x^2) + (49y^2) = 784 algebraically. Do not round. Find the correct ordered pairs of the foci.

Answer 1

To find the foci of an ellipse given by the equation (16x^2) + (49y^2) = 784, we first need to write the equation in standard form:

(16x^2) + (49y^2) = 784

Dividing by 784, we get:

(16x^2)/784 + (49y^2)/784 = 1
x^2/49 + y^2/16 = 1

Now we can identify the values of a and b for this ellipse, where a^2 is the denominator of the term with x^2 and b^2 is the denominator of the term with y^2. Therefore, a = 7 and b = 4.

The foci of an ellipse are given by the points (±c, 0), where c^2 = a^2 - b^2.

Substitute the values of a and b:

c^2 = 7^2 - 4^2
c^2 = 49 - 16
c^2 = 33
c = √33

Therefore, the foci are located at the points (-√33, 0) and (√33, 0).

The correct ordered pairs of the foci are:
Foci = (-√33, 0) and (√33, 0)