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The asymptotes of a hyperbola are \(y=2x-3\) and \(y=17-2x\) Also, the hyperbola passes through the point \((4,7)\) Find the distance between the foci of the hyperbola. Once again, I would greatly appreciate any help given. Thanks!

 Feb 5, 2020
edited by Impasta  Feb 6, 2020
 #1
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The distance between the foci is 2*sqrt(5).

 Feb 5, 2020
 #2
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The asymptotes of a hyperbola are y=2x-3 and y=17-2x. Also, the hyperbola passes through the point  Find the distance between the foci of the hyperbola. Once again, I would greatly appreciate any help given. Thanks!

laugh

 Feb 5, 2020
edited by Omi67  Feb 5, 2020
 #3
avatar+25237 
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The asymptotes of a hyperbola are \(y=2x-3\) and \(y=17-2x\)
Also, the hyperbola passes through the point \((x_p,\ y_p)\)
Find the distance between the foci of the hyperbola.


\(\text{The distance between the foci of the hyperbola $\mathbf{= \sqrt{k}\sqrt{5}} \\$ with $k = -(2x_p-3-y_p)(17-2x_p-y_p)$ }\)

 

The formula of the hyperbola:

\(\begin{array}{|rcll|} \hline \dfrac{(x-5)^2}{a^2}-\dfrac{(y-7)^2} {b^2} &=& 1 \\ a &=&\dfrac{\sqrt{k}}{2}\\ b &=& \sqrt{k} \\ k &=& -(2x_p-3-y_p)(17-2x_p-y_p) \\ \hline \end{array}\)

 

\(\begin{array}{|rcll|} \hline x_p = 4,\ y_p = 7 \\ \\ k &=& -(2x_p-3-y_p)(17-2x_p-y_p) \\ k &=& -(2*4-3-7)(17-2*4-7) \\ k &=& -(8-10)(10-8) \\ k &=& (10-8)(10-8) \\ k &=& 2*2 \\ \mathbf{k} &=& \mathbf{4} \\\\ a &=&\dfrac{\sqrt{4}}{2}\\ \mathbf{a} &=& \mathbf{1} \\\\ b &=& \sqrt{k} \\ b &=& \sqrt{4} \\ \mathbf{b} &=& \mathbf{2} \\\\ c^2 &=& a^2+b^2 \\ c^2 &=& 1^2+2^2 \\ c^2 &=& 5 \\ \mathbf{c} &=& \mathbf{\sqrt{5}} \\\\ \overline{F_1F_2} &=& 2c \\ \mathbf{\overline{F_1F_2}} &=& \mathbf{2\sqrt{5}} \\ \hline \end{array}\)

 

The distance between the foci of the hyperbola is \(\mathbf{2\sqrt{5}}\)

 

 

laugh

 Feb 5, 2020
edited by heureka  Feb 5, 2020
edited by heureka  Feb 5, 2020
edited by heureka  Feb 6, 2020
edited by heureka  Feb 6, 2020
edited by heureka  Feb 6, 2020
edited by heureka  Feb 6, 2020

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