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

1 Answer
Sep 25, 2014

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.