Question

Il. Analyze the hyperbola with center (1,4), focus (-2, 4), and vertex (0, 4). That is,t nd: a) the other vertex 2 4 b) the other two points c) the transverse axis I) the other focus (4,4) the equation of the graph -))2-04

Answer 1

The standard equation for a hyperbola with a horizontal transverse axis and center at (h,k) is (x-h)²/a² – (y-k)²/b² = 1. The distance between the vertices is 2a and the distance between the foci is 2c. Also, a²+b² = c² .

Here, the center, focus, and vertex all lie on the horizontal line y = 4( i.e. these points lie on a line parallel to the X-Axis ). Thus, the hyperbola has a horizontal transverse axis.

a).The vertices of the hyperbola are equidistant from the center so that the other vertex is at (2,4).

b). Which other 2 points?

c).The center, focus, and vertex all lie on the horizontal line y = 4 which is the transverse axis.

d). The focus is 3 units from the center, so c = 3. Hence, the other focus is (4,4).

e). The vertex is 1 unit from the center so that a = 1. Now, since a²+b² = c² , hence b² = 9-1 = 8. Thus, the equation of the hyperbola is (x-1)²/1 - (y-4)²/8 = 1.