To create a rotary the Department of Roads and Highways woul

To create a rotary, the Department of Roads and Highways would like to redesign the intersection of Washington Street and Lincoln Avenue so that the edges of the roads are in the shape of two hyperbolas. The hyperbolas are defined by using the existing roads as the asymptotes. Because of the placement of the fire station and the police station, the maximum horizontal distance of the rotary is 330 feet. A surveyor took measure of Washington Street and Lincoln Avenue. Lincoln Avenue’s path is in the ratio of 3 units west for each 2 units north. Washington Street’s path is in the ratio of 3 units east for each 2 units north. Find the equations of the two hyperbolas that form the edges of the new roads.

Solution

Given

the maximum horizontal distance of the rotary is 330 feet

so this distance should be equal to 2a

2a = 330

a = 165 feet

since the ratio is 3 : 2

3b = 2a

3b = 2*165

b = 330/3

b = 110 feet

now the equation of the hyperbola can be given as

x²/a² - y²/b² = 1   and

y²/b² - x²/a² = 1

we get

x²/(165)² - y²/(110)² = 1 and

y²/(110)² - x²/(165)² = 1