Question

Check MyAnswer Please!!!

What is the type of conic section given by the equation x^2 - 9y^2 = 900 and what is the domain and range?
My Answer: Hyperbola
Domanin: All real values ofx
Range:???
Please explain how to find the range and the domain

Answer 1

The equation
`x^2-9y^2=900` defines a hyperbolic cylinder.

1. Divide the equation
`x^2-9y^2=900` by 900:

`(x^2)/(900) - (y^2)/(100) =1`.
This is the equation of conic section when z=0 (or z=const) and that is hyperbola equation.

2. To find the domain and the range you should express y:

`(y^2)/(100) =(x^2)/(900) - 1 \\ \\ y^2= (x^2)/(9) -100 \\ \\ y=\pm \sqrt{(x^2)/(9) -100}`.
Since you have square root,

`(x^2)/(9) -100\ge 0 \\ x^2-900\ge 0 \\ (x-30)(x+30)\ge 0 \\ x\in(-\infty,-30)\cup (30,\infty)`. This is the domain, the range is
`y\in (-\infty,\infty)`.