Question

How do you classify the conic `4x^2-y^2-8x+4y-9=0`?

Answer 1

It is a hyperbola.

Explanation:

`4x^2-y^2-8x+4y-9=0`

`hArr4(x^2-2x+1)-4-(y^2+4y+4)+4-9=0`

or `4(x-1)^2-(y+2)^2=9`

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

As `4x^2-y^2-8x+4y-9=0` can be written as `(x-h)^2/a^2-(y-k)^2/b^2=1`, it is a hyperbola.

graph{4x^2-y^2-8x+4y-9=0 [-9.46, 10.54, -3.76, 6.24]}

Answer 2

The General Cartesian form for a conic section is:

`Ax^2+Bxy+Cy^2+Dx+Ey+F = 0`
One may use the value of `B^2-4AC` to classify it.

Explanation:

The given equation

`4x^2-y^2-8x+4y-9=0`

Fits the General Cartesian form with `A = 4, B, = 0, C = -1, D = -8, E = 4 and F=-9`

Refering to the reference section entitled, Discriminant , we compute the value:

`B^2 -4AC = 0^2-4(4)(-1) = 16`

The value is greater than zero. The list of conditions given by reference tells us that the conic section is a hyperbola.