Use the eccentricity to find the equation of the hyperbola in standard form. vertices (1, 6)and (1, 4), eccentricity 2

Question

Quizplus · Accepted Answer

The center of the hyperbola is halfway between the vertices, so it is at (1,5). The distance between the vertices is 2a = 2, so a = 1. The eccentricity is e = 2, so c = ae = 2. Since the transverse axis is vertical, the equation of the hyperbola in standard form is:

(y - 5)^2 / b^2 - (x - 1)^2 / a^2 = 1

Plugging in the values of a, c, and the vertex, we get:

(y - 5)^2 / b^2 - (x - 1)^2 / 1^2 = 1
(y - 5)^2 / b^2 - (x - 1)^2 = 1
(y - 5)^2 / b^2 = (x - 1)^2 + 1

To find b^2, we use the formula e^2 = c^2 - a^2:

2^2 = 2^2 - 1^2
b^2 = 3

Therefore, the equation of the hyperbola in standard form is:

(y - 5)^2 / 3 - (x - 1)^2 / 1 = 1

Use the Eccentricity to Find the Equation of the Hyperbola