+26393 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 \(\mathbf{2\sqrt{5}}\)
