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 and showing correct, step-by-step work
First, we need to put the given equation in standard form for an ellipse by dividing both sides by 784:
(16x^2)/784 + (49y^2)/784 = 1
x^2/49 + y^2/16 = 1
Next, we can identify the values of a and b where a = √49 = 7 and b = √16 = 4.
The foci for an ellipse are given by the formula c = √(a^2 - b^2).
Plugging in the values of a and b, we get:
c = √(7^2 - 4^2) = √(49-16) = √33
Now, to find the foci, we need to determine if 33 is under the x or y term. Since:
a^2 = b^2 + c^2
49 = 16 + 33
This implies the foci are along the x-axis. Therefore, the foci are at the points (±√33, 0).
So, the foci of the ellipse are (±√33, 0).