A hyperbola centered at the origin has a vertex at $(0,-40)$ and a focus at $(0,41)$.
\begin{tabular}{|l|l|}
\hline Vertices: (-a,0), (a,0) & Vertices: ( $0,-a),(0, a)$ \\
\hline Foci: (-c,0),(c,0) & Focii $(0,-c),(0, c)$ \\
\hline Asymptotes: $y= \pm \frac{b}{a} x$ & Asymptotes: $y= \pm \frac{a}{b} x$ \\
\hline Directrices: $x= \pm \frac{a^2}{c}$ & Directrices: $y= \pm \frac{a^2}{c}$ \\
\hline Standard Equation: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ & Standard Equation: $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ \\
\hline
\end{tabular}
Which are the equations of the asymptotes?
Answer (2)
Identifies the hyperbola's parameters: vertex ( 0 , − 40 ) implies a = 40 , focus ( 0 , 41 ) implies c = 41 .
Applies the hyperbola relationship c 2 = a 2 + b 2 to find b 2 = 4 1 2 − 4 0 2 = 81 , thus b = 9 .
Uses the formula for asymptotes y = ± b a x with a = 40 and b = 9 .
Concludes that the equations of the asymptotes are y = ± 9 40 x .
Explanation
Analyze the problem and given data We are given a hyperbola centered at the origin with a vertex at ( 0 , − 40 ) and a focus at ( 0 , 41 ) . We need to find the equations of the asymptotes. From the table provided, since the vertices and foci are on the y-axis, the standard equation of the hyperbola is of the form a 2 y 2 − b 2 x 2 = 1 , and the asymptotes are given by y = ± b a x .
Find the values of a and c Since the vertex is at ( 0 , − 40 ) , we have a = 40 . Since the focus is at ( 0 , 41 ) , we have c = 41 . For a hyperbola, the relationship between a , b , and c is given by c 2 = a 2 + b 2 .
Calculate b^2 Substitute the values of a and c into the equation c 2 = a 2 + b 2 to find b 2 :
4 1 2 = 4 0 2 + b 2
Solve for b 2 :
b 2 = 4 1 2 − 4 0 2 = 1681 − 1600 = 81
Calculate b Therefore, b = 81 = 9 .
Find the equations of the asymptotes The equations of the asymptotes are y = ± b a x = ± 9 40 x . Thus, the equations of the asymptotes are y = 9 40 x and y = − 9 40 x .
Examples
Understanding hyperbolas and their asymptotes is crucial in various fields, such as physics and engineering. For instance, the trajectory of a comet as it approaches and recedes from the sun can be modeled as a hyperbola, with the asymptotes providing an approximation of the comet's path far from the sun. Similarly, in radio navigation systems like LORAN, hyperbolas are used to determine the location of a receiver based on the time difference of arrival of signals from different transmitters. The asymptotes help in understanding the boundaries of the possible locations.
The equations of the asymptotes for the hyperbola centered at the origin with a vertex at (0, -40) and a focus at (0, 41) are y = 9 40 x and y = − 9 40 x . To derive these equations, we identified the values of a and b and utilized the formula for asymptotes of a hyperbola with a vertical transverse axis.
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