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.

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

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).