What conic section does the equation x2+4y2−4x+8y−60=0 represent?

1 Answer
AJ Speller
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.

x2−4x+4y2+8y=60

Let's work with the x term

(−42)2=(−2)2=4, We need to add 4 to both sides of the equation

x2−4x+4+4y2+8y=60+4

x2−4x+4⇒(x−2)2⇒Perfect square trinomial

Re-write equation:

(x−2)2+4y2+8y=60+4

Let's factor out a 4 from the y2 & y terms

(x−2)2+4(y2+2y)=60+4

Let's work with the y term

(22)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(y2+2y+1)=60+4+4

y2+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)268+4(y+1)268=6868

(x−2)268+(y+1)217=1

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

The x-axis is the major axis.

The y-axis is the minor axis.

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