Answer to Question #278479 in Civil and Environmental Engineering for Ling

Question

Find the equation of the hyperbola with center (4, - 1), transverse axis parallel to the Y-axis, distance between foci 10, latus rectum 9/2.

Answer 1

Latus rectum is 2p

And p = b^2/a ------------------------------------- (1)

Distance between foci is 2c = 2√(145)

And b^2 = c^2 – a^2  ---------------------------- (2)

Slope of asymptote is |m| = b/a

And m = (1/16)*(2p)  (using positive slope)

b/a = p/8   ------------------------------------------ (3)   

Using (3) and (1)

8b/a = p = b^2/a

b = 8

Using (2) with c = √(145),

64 = 145 – a^2

a^2 = 81

a = 9

Eqn of Hyperbola

x^2/a^2 – y^2/b^2 = 1

x^2/81 – y^2/64 = 1       (Centred at (0,0))

(x – 2)^2/81 – (y + 5)^2/64 = 1        (Centred at (2, -5))