Find a Cartesian equation for the hyperbola which has asymptotes
and which passes through the origin.
Incomplete.
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One comment
S says:
We can assume that when we shift the asymptotes to (0,0), we will have the hyperbola discussed in section 13.22. Such asymptotes are y = +/-(2x). From that we know that a/b=2 and y^2/a^2 – x^2/b^2 = 1. The shifted hyperbola to those asymptotes goes through (-1/2, -2). We plug in the values and get (y-2)^2 – (2x-1)^2=3, which is the solution in the book. (Sorry, I noticed that I skipped one excercise and the comments got out of sync with the text…)
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![\[ y = 2x + 1 \qquad \text{and} \qquad y = -2x + 3 \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-1338d67e00507090ea1c5858ff5d9870_l3.png "Rendered by QuickLaTeX.com")
We can assume that when we shift the asymptotes to (0,0), we will have the hyperbola discussed in section 13.22. Such asymptotes are y = +/-(2x). From that we know that a/b=2 and y^2/a^2 – x^2/b^2 = 1. The shifted hyperbola to those asymptotes goes through (-1/2, -2). We plug in the values and get (y-2)^2 – (2x-1)^2=3, which is the solution in the book. (Sorry, I noticed that I skipped one excercise and the comments got out of sync with the text…)