Question

What is the type of conic section given by the equation x^2 - 9y^2 = 900 and what is the domain and range?

Answer 1

A conic section is a curve obtained by the intersection of the surface of a cone with a plane. A conic section can be a circle, a hyperbola, a parabola, and an ellipse.

For a circle, the general equation of a circle with center, (a, b), and a radius, r, is of the form
$(x-a)^2+(y-b)^2=r^2$

For a hyperbola, the general equation of a hyperbola with center (h, k), and a and b half the lengths of the major and the minor axis respectively is of the form.
$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} =1$

For a parabola, the general equation of a parabola with center (h, k), and a multiplier a is of the form
$y-k=a(x-h)^2$

For an ellipse, the general equation of an ellipse with center (h, k), and a and b half the lengths of the major and the minor axis respectively.
$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} =1$

Given the equation
$x^2-9y^2=900$
It can be rewritten as
$\frac{(x-0)^2}{900} - \frac{(y-0)^2}{100} =1 \\ \\ \frac{(x-0)^2}{30^2} - \frac{(y-0)^2}{10^2} =1$
This gives an equation of a hyperbola with center (0, 0), half the length of the major axis = 30 and half the length of the minor segment = 10.

The domain of the equation is all real values of x.