Question

How do you write the hyperbola `y^2-x^2/36=4` in standard form?

Answer 1

`color(green)((y-0)^2/(2^2)-(x-0)^2/(12^2)=1`

Explanation:

The standard form for a hyperbola is
either
`color(white)("XXX")(x-h)^2/(a^2)-(y-k)^2/(b^2)=1` for a hyperbola opening left-right
or
`color(white)("XXX")(y-k)^2/(b^2)-(x-a)^2/(a^2)=1` for a hyperbola opening up-down.
In both cases the center is at `(h,k)`

Given
`color(white)("XXX")y^2-x^2/36=4`
we obviously will need to transform this into the second (up-down opening) form.

`color(white)("XXX")y^2/4-x^2/(4 * 36) =1`

`color(white)("XXX")y^2/2^2 - x^2/(12^2)=1`

or (if you wish to make the center explicit)
`color(white)("XXX")(y-0)^2/(2^2)-(x-0)^2/(12^2)=1`

While, not technically in "standard form", this could be simplified as:
`color(white)("XXX")y^2/4-x^2/144=1`