Question

What conic section does the equation `x^2 + 4y^2 - 4x + 8y - 60 = 0` represent?

Answer 1

In this problem we are going to rely on the completing the square technique to massage this equation into an equation that is more recognizable.

`x^2-4x+4y^2+8y=60`

Let's work with the `x` term

`(-4/2)^2=(-2)^2=4`, We need to add 4 to both sides of the equation

`x^2-4x+4+4y^2+8y=60+4`

`x^2-4x+4 => (x-2)^2 =>`Perfect square trinomial

Re-write equation:

`(x-2)^2+4y^2+8y=60+4`

Let's factor out a 4 from the `y^2` & `y` terms

`(x-2)^2+4(y^2+2y)=60+4`

Let's work with the `y` term

`(2/2)^2=(1)^2=1`, We need to add 1 to both sides of the equation

But remember that we factored out a 4 from the left side of the equation. So on the right side we are actually going to add 4 because `4*1=4.`

`(x-2)^2+4(y^2+2y+1)=60+4+4`

`y^2+2y+1 => (y+1)^2 =>`Perfect square trinomial

Re-write equation:

`(x-2)^2+4(y+1)^2=60+4+4`

`(x-2)^2+4(y+1)^2=68`

`((x-2)^2)/68+(4(y+1)^2)/68=68/68`

`((x-2)^2)/68+((y+1)^2)/17=1`

This is an ellipse when a center (2,-1).

The `x`-axis is the major axis.

The `y`-axis is the minor axis.