jmc

Question

Let F_1 = (10,2) and F_ 2= (-16,2). Then the set of points P such that |PF_1 - PF_2| = 24form a hyperbola. The equation of this hyperbola can be written as (x - h)^2a^2 - (y - k)^2b^2 = 1.Find h + k + a + b.

Accepted Answer

The center of the hyperbola is the midpoint of

\overline{F_{1} F_{2}},

which is

(- 3, 2) .

Thus,

h = - 3

and

k = 2.

Also,

2 a = 24,

so

a = 12.

The distance between the foci is

2 c = 26,

so

c = 13.

Then

b^{2} = c^{2} - a^{2} = 169 - 144 = 25,

so

b = 5.

Hence,

h + k + a + b = (- 3) + 2 + 12 + 5 = 16 .