Use a suitable rotation of axes to find an equation for the

Use a suitable rotation of axes to find an equation for the graph in an x\'y\' plane, and sketch the graph, labeling vertices. x²- 2xy + y² + 4x + 4y = 0

Solution

x^2-2xy+y^2+4x+4y=0

(x-y)^2+4(x+y)=0

Now the lines:x+y=0,x-y=0 are perpendicular lines passing through the origin. And wih the change of variables:

x\'=x+y,y\'=x-y the equation becomes

x\'^2+4y\'=0

x+y=0 makes angle: 45 degrees with x axis

So we need a 45 degree rotation of x and y axis counter clockwise