Problem 71 - College Algebra

Question

Describe how to locate the foci for \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\)

Step-by-Step Solution

Answer

The foci for the given equation \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\) are located at (-3,0) and (3,0)

Step 1: Identify the Major and Minor Axes

Looking at the equation \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\), we can see that the major axis 'a' is given by \(\sqrt{25} = 5\) and minor axis 'b' is given by \(\sqrt{16} = 4\). The larger denominator is the square of the major axis, and the smaller is the square of the minor axis.

Step 2: Calculate the measure of the foci

Then use the formula \(c = \sqrt{a^2 - b^2}\) to calculate the distance from the center to a focus point. When we input the values for a and b, we got \(c = \sqrt{5^2 - 4^2} = \sqrt{9} = 3\)

Step 3: Provide the foci coordinates

Finally, note that the foci will be located along the major axis, which in this case is along the x-axis. Here, the center of the ellipse is at the origin (0,0). So foci points will be 3 units left and right from the center. Consequently, the coordinates of the foci are (-3,0) and (3,0).