Answer to Question #147448 in Analytic Geometry for Nicole tenorio

Question

An engineer is working on making a diagram for the cooling tower he was going to construct in the future. Cooling towers can be seen as towers with inward curving sides, or simply, hyperbola looking sides. His daughter asks him how high the center of the hyperbolic sides of tower was y=±(24/7)x+15 and the answers that the asymptotes of the hyperbola are the following, and the area of itsa auxilliary rectangle is 168 squats units. The daughter then find the foci of the hyperbola. At what points on the cartesian plane are these foci found

Answer 1

$\text{Find the point O of intersection of the asymptotes}$

$\frac{24}{7}x+15 = -\frac{24}{7}x+15;x_0= 0 \ y_0=15$

$\text{The hyperbola equation has the form:}$

$\frac{x^2}{a^2}-\frac{(y-15)^2}{b^2}=1$

$\frac{b}{a}=\frac{24}{7}\text{ from the equation of asymptotes}$

$2*b*2*a = 168$ $\text{from the area of the construction rectangle}$

$\text{from here } b=12;a=3.5$

$c = \sqrt{a^2+b^2}=12.5$

$F_1(12.5,15)\ F_2(-12.5,15) -\text{focus coordinates}$

Answer:$F_1(12.5,15)\ F_2(-12.5,15) -\text{focus coordinates}$