Problem 73 - College Algebra

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 ellipses represented by the two equations are similar in their shape and size due to the same semi-axes length (a=5, b=4). The difference is in their locations or centers on the coordinate plane, with the first centered at (0,0) and the second at (1,1).

Step 1: Understand the Equations

Let's first understand the nature of the equations. Both equations are of the form \(\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1\), which describes an ellipse centered at (h, k). The a and b values denote the length of the semi-axes of the ellipse.

Step 2: Analyzing the first Equation

The ellipse of the first equation \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\) has its center at origin (0,0) because there are no constant values (h,k) subtracted in its form.

Step 3: Analyzing the second Equation

In the second equation \(\frac{(x-1)^{2}}{25}+\frac{(y-1)^{2}}{16}=1\), the ellipse is centered at (1,1), which means it has undergone a shift on the plane.

Step 4: The Similarity and Difference

The similarity is that both equations represent ellipses with the same length of semi-axes (a=5, b=4), meaning they have the same size and shape. The difference, however, is their location on the coordinate plane. The first ellipse is centered at the origin (0,0) while the second ellipse is centered at (1,1). Hence, the second ellipse is a horizontal and a vertical shift of the first one.