Multiple Choice
Find the equation for a hyperbola with a center at , focus at and vertex at .
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8. Conic Sections
Hyperbolas at the Origin
5:37 minutesTextbook Question
Graph the hyperbola. Locate the foci and find the equations of the asymptotes. (x^2)/16 - y^2 = 1
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Step 1: Recognize the standard form of the hyperbola. The given equation \( \frac{x^2}{16} - y^2 = 1 \) is in the standard form of a hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), where the hyperbola opens left and right because the \( x^2 \) term is positive.
Step 2: Identify the values of \( a^2 \) and \( b^2 \). From the equation, \( a^2 = 16 \) (so \( a = 4 \)) and \( b^2 = 1 \) (so \( b = 1 \)). These values will help determine the vertices, foci, and asymptotes.
Step 3: Locate the vertices. The vertices of the hyperbola are located at \( (\pm a, 0) \), which means the vertices are at \( (4, 0) \) and \( (-4, 0) \).
Step 4: Find the foci. The distance from the center to each focus is given by \( c = \sqrt{a^2 + b^2} \). Substitute \( a^2 = 16 \) and \( b^2 = 1 \) to calculate \( c \). The foci are located at \( (\pm c, 0) \).
Step 5: Write the equations of the asymptotes. The asymptotes of a hyperbola in this form are given by \( y = \pm \frac{b}{a}x \). Substitute \( b = 1 \) and \( a = 4 \) to find the equations of the asymptotes.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Hyperbola Definition
A hyperbola is a type of conic section formed by the intersection of a plane and a double cone. It consists of two separate curves called branches, which are mirror images of each other. The standard form of a hyperbola can be expressed as (x^2/a^2) - (y^2/b^2) = 1 for horizontal hyperbolas, where 'a' and 'b' determine the shape and size of the hyperbola.
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Foci of a Hyperbola
The foci of a hyperbola are two fixed points located along the transverse axis, which is the line that passes through the center and the vertices of the hyperbola. For the hyperbola given by (x^2/a^2) - (y^2/b^2) = 1, the distance from the center to each focus is calculated using the formula c = √(a^2 + b^2). The foci play a crucial role in defining the hyperbola's shape and properties.
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Asymptotes of a Hyperbola
Asymptotes are straight lines that the branches of a hyperbola approach but never touch. For the hyperbola in the form (x^2/a^2) - (y^2/b^2) = 1, the equations of the asymptotes can be derived as y = ±(b/a)x. These lines provide a framework for understanding the behavior of the hyperbola at extreme values and are essential for accurate graphing.
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