Question

Describe one similarity and one difference between the graphs of x2/25 + y2/16 = 1 and (x - 1)2/25 + (y - 1)2/16 = 1.

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asked 2020-12-05
Describe one similarity and one difference between the graphs of \(\displaystyle{x}\frac{{2}}{{25}}+{y}\frac{{2}}{{16}}={1}{\quad\text{and}\quad}{\left({x}-{1}\right)}\frac{{2}}{{25}}+{\left({y}-{1}\right)}\frac{{2}}{{16}}={1}\).

Answers (1)

2020-12-06
Step 1
Consider the following equations of two ellipses, \(\displaystyle\frac{{x}^{{2}}}{{25}}+\frac{{y}^{{2}}}{{16}}={1}{\quad\text{and}\quad}\frac{{{\left({x}-{1}\right)}^{{2}}}}{{{25}}}+\frac{{{\left({y}-{1}\right)}^{{2}}}}{{16}}={1}\)
Step 2
In the given equations \(\displaystyle\frac{{x}^{{2}}}{{25}}+\frac{{y}^{{2}}}{{16}}={1}{\quad\text{and}\quad}\frac{{{\left({x}-{1}\right)}^{{2}}}}{{{25}}}+\frac{{{\left({y}-{1}\right)}^{{2}}}}{{16}}={1}\), he denominator of the \(\displaystyle{x}^{{2}}\) term is greater than the denominator of the \(\displaystyle{y}^{{2}}\)-term, so the major axis horizontal and parallel to x-aixs.
Compare the first equation \(\displaystyle\frac{{x}^{{2}}}{{25}}+\frac{{y}^{{2}}}{{16}}={1}\) with the standard form \(\displaystyle\frac{{{\left({x}-{h}\right)}^{{2}}}}{{a}^{{2}}}+\frac{{{\left({y}-{k}\right)}^{{2}}}}{{b}^{{2}}}={1}\).
It is observed that \(\displaystyle{a}^{{2}}={25},{b}^{{2}}={16},{h}={0}{\quad\text{and}\quad}{k}={0}\).
That is, a = 5, b = 4, h = 0 and k = 0.
The center of the ellipse \(\displaystyle\frac{{x}^{{2}}}{{25}}+\frac{{y}^{{2}}}{{16}}={1}\) is at origin, major axis is along x-axis and minor axis is along y-axis.
Compare the second equation \(\displaystyle\frac{{{\left({x}-{1}\right)}^{{2}}}}{{{25}}}+\frac{{{\left({y}-{1}\right)}^{{2}}}}{{16}}={1}\) with the standard form \(\displaystyle\frac{{{\left({x}-{h}\right)}^{{2}}}}{{a}^{{2}}}+\frac{{{\left({y}-{k}\right)}^{{2}}}}{{b}^{{2}}}={1}\)
It is observed that \(\displaystyle{a}^{{2}}={25},{b}^{{2}}={16}\), h = 1 and k = 1.
That is, a = 5, b = 4, h = 1 and k = 1.
The center of the ellipse \(\displaystyle\frac{{{\left({x}-{1}\right)}^{{2}}}}{{{25}}}+\frac{{{\left({y}-{1}\right)}^{{2}}}}{{16}}={1}\) is at point (1,1), major axis is along x-axis and minor axis is along y-axis.
Similarity is the major axis of two graphs is horizontal and of same length. Difference is center of ellipse
\(\displaystyle\frac{{x}^{{2}}}{{25}}+\frac{{y}^{{2}}}{{16}}={1}\) is at (0,0) and center of ellipse \(\displaystyle\frac{{{\left({x}-{1}\right)}^{{2}}}}{{{25}}}+\frac{{{\left({y}-{1}\right)}^{{2}}}}{{16}}={1}\) is at point (1,1).
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