Problem 75 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The foci of the given ellipse are at (-3, 0) and (3, 0).

Step 1: Determine the major and minor axes

Comparing the given equation \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\) with the standard equation of ellipse, we can see that 'a' is 5 and 'b' is 4. Since 'a' is greater than 'b', 'a' is the semi major-axis and 'b' is the semi-minor axis.

Step 2: Calculate the distance 'c'

Using the formula \(c = \sqrt{a^{2} - b^{2}}\), we can substitute 'a' as 5 and 'b' as 4 to get \(c = \sqrt{5^{2} - 4^{2}} = \sqrt{9} = 3\)

Step 3: Locate the foci

Since the the greater denominator is under \(x^{2}\), the foci are on the x-axis, 'c' units away from origin. Therefore the coordinates of the foci are (-3, 0) and (3, 0).