# Describe one similarity and one difference between the graphs of x^2/9 - y^2/1 = 1 and ((x - 3)^2)/9 - ((y + 3)^2)/1 = 1.

Question
Similarity
Describe one similarity and one difference between the graphs of $$\displaystyle\frac{{x}^{{2}}}{{9}}-\frac{{y}^{{2}}}{{1}}={1}{\quad\text{and}\quad}\frac{{{\left({x}-{3}\right)}^{{2}}}}{{9}}-\frac{{{\left({y}+{3}\right)}^{{2}}}}{{1}}={1}$$.

2020-10-27
Given:
(a).$$\displaystyle\frac{{x}^{{2}}}{{9}}-\frac{{y}^{{2}}}{{1}}={1}$$
(b).$$\displaystyle\frac{{{\left({x}-{3}\right)}^{{2}}}}{{9}}-\frac{{{\left({y}+{3}\right)}^{{2}}}}{{1}}={1}$$
To describe: One similarity and one difference between both the graphs.
Concept:
The sandard form of the equation of a hyperbola with center
(h,k)and transverse axis parallel to x - axis is
$$\displaystyle\frac{{{\left({x}-{h}\right)}^{{2}}}}{{a}^{{2}}}-\frac{{{\left({y}-{k}\right)}^{{2}}}}{{b}^{{2}}}={1}$$, then
Lenght of transverse axis 2a.
Lenght of conjugate axis is 2b.
Distance between the foci is 2c, where $$\displaystyle{c}^{{2}}={a}^{{2}}+{b}^{{2}}$$.
Step 2
Explanation:
Here we have $$\displaystyle\frac{{x}^{{2}}}{{9}}-\frac{{y}^{{2}}}{{1}}={1}{\quad\text{and}\quad}\frac{{{\left({x}-{3}\right)}^{{2}}}}{{9}}-\frac{{{\left({y}+{3}\right)}^{{2}}}}{{1}}={1}$$
i.e. $$\displaystyle\frac{{x}^{{2}}}{{{\left({3}\right)}^{{2}}}}-\frac{{y}^{{2}}}{{{\left({1}\right)}^{{2}}}}={1}{\quad\text{and}\quad}\frac{{{\left({x}-{3}\right)}^{{2}}}}{{{\left({3}\right)}^{{2}}}}-\frac{{{\left({y}+{3}\right)}^{{2}}}}{{{\left({1}\right)}^{{2}}}}={1}$$
Difference: Both the hyperbola have different centre.
Center of $$\displaystyle\frac{{x}^{{2}}}{{{\left({3}\right)}^{{2}}}}-\frac{{y}^{{2}}}{{{\left({1}\right)}^{{2}}}}={1}$$ is (0,0) while center of $$\displaystyle\frac{{{\left({x}-{3}\right)}^{{2}}}}{{{\left({3}\right)}^{{2}}}}-\frac{{{\left({y}+{3}\right)}^{{2}}}}{{{\left({1}\right)}^{{2}}}}={1}$$ is (3,-3).
Similarity:Since both the hyperbola's have the same value of $$\displaystyle\alpha={3}{\quad\text{and}\quad}{b}={1}$$
$$\displaystyle\Rightarrow$$ Both the parabola's have same
lenght of transverse axes = 2a
lenght of conjugate axes = 2b
and distance between the foci = 2c.

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