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Chapter 13: Problem 50

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
Both ellipses have the same shape, orientation, and size, but are located at different places on the coordinate plane.

Step by step solution

01

Identify the properties of the first ellipse

The equation \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\) represents an ellipse with its center at the origin (0,0), a horizontal radius of 5 and a vertical radius of 4.
02

Identify the properties of the second ellipse

The equation \(\frac{(x-1)^{2}}{25}+\frac{(y-1)^{2}}{16}=1\) also represents an ellipse. The center of this ellipse is at the point (1,1), and it also has a horizontal radius of 5 and a vertical radius of 4.
03

Compare the properties of the ellipses

Both ellipses have the same shape and size (as they have the same horizontal and vertical radii), so this is a similarity. However, they have different centers. The first ellipse is centered on the origin, while the second ellipse is shifted one unit to the right and one unit up, to the point (1,1). This is a difference between the graphs.

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Most popular questions from this chapter

If you are given an equation of a parabola, explain how to determine if the parabola opens to the right, to the left, upward, or downward.

The equation of a parabola is given. Determine: a. if the parabola is horizontal or vertical. b. the way the parabola opens. c. the vertex. $$x=-(y+3)^{2}+4$$

Use the vertex and intercepts to sketch the graph of each equation. If needed, find additional points on the parabola by choosing values of y on each side of the axis of symmetry. $$x=-(y-3)^{2}+4$$

How can you distinguish ellipses from circles by looking at their equations?

Use the vertex and intercepts to sketch the graph of each equation. If needed, find additional points on the parabola by choosing values of y on each side of the axis of symmetry. $$x=(y-4)^{2}+1$$
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