Question

Is the following equation an ellipse, circle, parabola, or hyperbola `4x^2-3y^2-48x-6y+129=0`?

Answer 1

The reference Conic Section tells us that given equation:

`4x^2-3y^2-48x-6y+129=0`

Is in the General Cartesian form:

`Ax^2+Bxy+Cy^2+Dx+Ey+F=0`

where `A = 3, B= 0, and C =-3`

The same reference, also. tells us that the determinant, `B^2-4AC` and be used to determine the type of conic section:

Compute the determinant:

`B^2-4AC = 0^2-4(3)(-3) = 36`

The determinant is greater than zero, therefore, the conic section is a hyperbola. Here is its graph as proof.

graph{4x^2-3y^2-48x-6y+129=0 [-7.37, 21.11, -7.57, 6.67]}