Problem 44 - College Algebra

Question

Graph each ellipse and give the location of its foci. $$\frac{(x-4)^{2}}{4}+\frac{y^{2}}{25}=1$$

Step-by-Step Solution

Answer

The foci of the ellipse are located at (4, \(\sqrt{21}\)) and (4, -\(\sqrt{21}\)). The semi-minor axis is 2 and the semi-major axis is 5 with the center of the ellipse at (4, 0).

Step 1: Identify the Ellipse Parameters

In the given equation \((x-4)^{2}/4 + y^{2}/25 = 1\), the coordinates of the center (h, k) can be found as h = 4 and k = 0 because the equation is in the form \((x-h)^2/a^2 + (y-k)^{2}/b^2 = 1\), here a^2 = 4 so a = 2, and b^2 = 25 so b = 5. Thus, the center is (4, 0), the semi-minor axis is 2 and the semi-major axis is 5.

Step 2: Find the Foci

Using the relationship for the foci \(c= \sqrt{|a^{2}-b^{2} |}\), we find \(c = \sqrt{|2^{2} - 5^{2}|} = \sqrt{21}\) since \(-21\) under a sqrt sign yields an imaginary result.

Step 3: Plot the Ellipse

Using the center (4, 0), points along the semi-major axis at (4, 5) and (4, -5), and points along the semi-minor axis at (2, 0) and (6, 0), we can sketch the ellipse. The foci are at \((4, \sqrt{21})\) and \((4, -\sqrt{21})\)