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Chapter 7: Problem 73

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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.
04

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.

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Most popular questions from this chapter

Graph \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) and \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1\) in the same viewing rectangle for values of \(a^{2}\) and \(b^{2}\) of your choice. Describe the relationship between the two graphs.

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: \((2,4) ;\) Directrix: \(x=-4\)

If you are given the standard form of the equation of a parabola with vertex at the origin, explain how to determine if the parabola opens to the right, left, upward, or downward.

Find the focus and directrix of the parabola with the given equation. Then graph the parabola. $$8 y^{2}+4 x=0$$

Describe one similarity and one difference between the graphs of \(y^{2}=4 x\) and \((y-1)^{2}=4(x-1)\)
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