Problem 77 - Algebra and Trigonometry

Question

Describe one similarity and one difference between the graphs of \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\) and \(\frac{(x-1)^{2}}{25}+\frac{(y-1)^{2}}{16}=1\)

Step-by-Step Solution

Answer

The two ellipses are similar in that they have the same size and shape, due to identical divisions by 25 and 16 in the equations. They differ in their location, as the center of the second ellipse is shifted to the point (1,1) while the first is at the origin (0,0).

Step 1 - Equation Analysis and Similarity

Examine the structure of both equations. The similar form of the equations, \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\) and \(\frac{(x-1)^{2}}{25}+\frac{(y-1)^{2}}{16}=1\), suggests that the graphs will both be those of ellipses. The division by 25 and 16 in both cases shows that the two ellipses will have the same major and minor axes lengths. This indicates that the ellipses are the same size and shape, providing a similarity between the two.

Step 2 - Difference in the Ellipses

The critical difference between the two ellipses lies in the introduction of the \(x-1\) and \(y-1\) terms in the second equation. These terms signify a shift in the location of the center of the ellipse. The original ellipse is centered at the origin (0,0), while the shifted ellipse is centered at the point (1,1). This shift constitutes a significant difference between the two resultant graphs.