How do you find the equation of the hyperbola with center at the origin and foci #(+-10,0)# and asymptotes #y=+-3/4x#?

Because the foci are horizontally oriented, #(-10, 0) and (10,0)#, we know that the hyperbola is the horizontal transverse axis type and its equation is:

#(x - h)^2/a^2 - (y - k)^2/b^2 = 1" [1]"#

We are told that the center is at the origin, therefore, both h and k equal zero; this makes equation [1] become equation [2]:

#(x - 0)^2/a^2 - (y - 0)^2/b^2 = 1" [2]"#

The reference tells us that the equations of the asymptotes are, #y = -b/a(x - h) + k and y = b/a(x - h) + k#

Matching the above with the given equations, #y = -3/4x and y = 3/4x#, we observe that:

#b/a = 3/4#

Which can be written as:

#b = 3/4a" [3]"#

Because the foci are a distance of 10 from the origin, we can write the following equation:

#10 = sqrt(a^2 + b^2)" [4]"#

Substitute equation [3] into equation [4] and then solve for "a":

#10 = sqrt(a^2 + (3/4a)^2)#

#100 = a^2 + 9/16a^2#

#100 = 25/16a^2#

#a^2 = 64#

#a = 8#

Use equation [3] to compute the value of b:

#b = 3/4(8)#

#b = 6#

This makes equation [2] become equation [5]:

#(x - 0)^2/8^2 - (y - 0)^2/6^2 = 1" [5]"#

Equation [5] is the answer.