QQuestionMathematics
QuestionMathematics
Find the equation of a hyperbola with foci (0,plus or minus^6) and the length of the transverse axis 4. Graph the hyperbola.
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Answer
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Step 1:: Write down the standard equation for a hyperbola.
where $2a$ is the length of the transverse axis and $2c$ are the foci.
The standard equation for a hyperbola centered at the origin is:
Step 2:: Find $a$ and $c$.
We are also given that the foci are at $(0, \pm 6)$, so $c = 6$.
Step 3:: Find $b^1$.
b^2 = c^2 - a^2 = 6^2 - 2^2 = 36 - 4 = 32
Step 4:: Write down the equation of the hyperbola.
\boxed{\frac{x^2}{4} - \frac{y^2}{32} = 1}
Simplifying, we get:
Step 5:: Graph the hyperbola.
Other convenient points include $(\pm 1, \pm 4)$ and $(\pm 2, \pm 8)$.
To graph the hyperbola, we can first plot the asymptotes, which are given by the equations: We then plot points on the hyperbola by finding the x- and y-intercepts and any other convenient points. Connecting these points and drawing the asymptotes, we get the graph of the hyperbola: 
Final Answer
The equation of the hyperbola is \frac{x^2}{4} - \frac{y^2}{32} = 1. The graph of the hyperbola is shown above.
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