Question

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.

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asked 2020-10-26
Describe one similarity and one difference between the graphs of \(\displaystyle\frac{{x}^{{2}}}{{9}}-\frac{{y}^{{2}}}{{1}}={1}{\quad\text{and}\quad}\frac{{{\left({x}-{3}\right)}^{{2}}}}{{9}}-\frac{{{\left({y}+{3}\right)}^{{2}}}}{{1}}={1}\).

Answers (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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