Question

The transverse axis is y =3, the conjugate axis is x = −6, the ratio of the length of the conjugate axis to the length of the transverse axis is 1 : 2, and one of the vertices is on the y ‑axis. Find the equation of the hyperbola.

Answer 1

The equation of the hyperbola is
`((x+6)^2)/(k^2) - ((y-3)^2)/(k^2) =1`.

We know that the transverse axis has length 2a and the conjugate axis has length 2b.

Since the ratio of the length of the conjugate axis to the length of the transverse axis is 1:2, we have 2b=2a. Therefore, a=b.

The center of the hyperbola is the intersection of the transverse axis and conjugate axis.

Since the transverse axis is y = 3 and the conjugate axis is x = -6, the center of the hyperbola is (-6, 3).

One of the vertices is on the y-axis, which means the other vertex is also on the y-axis.

Since the distance between a vertex and the center is a, we know that the vertices are (-6, 3 + a) and (-6, 3 - a).

Plugging in what we know, we get that the equation of the hyperbola is

`((x+6)^2)/(a^2) - ((y-3)^2)/(a^2) =1`

We can simplify this equation by letting k^2 =a^2, so the equation becomes
`((x+6)^2)/(k^2) - ((y-3)^2)/(k^2) =1`.

Therefore, the equation of the hyperbola is
`((x+6)^2)/(k^2) - ((y-3)^2)/(k^2) =1`