What is the graph of 3x^2−2sqrt3xy+y^2+2x+2sqrt3y=0, given cot2θ=(A−C)/B?

1 Answer
May 6, 2015

The presense of a cross product term (xy) indicates that this conic section has been rotated. The angle of rotation is found by using the given formula.

In this question: A=3, B=-2sqrt3 and C=1, so

cot 2 theta = (3-1)/(-2sqrt3) = -1/sqrt3

Therefore, 2 theta = 150^@ and theta = 75^@

In order to do the rotation substitution, we'll need sin theta and cos theta.

Use cos 2 theta = -sqrt3/2 and the half angle formulas to get sin theta and cos theta.

If you have not learned to use the discriminant, you'll need to rotate the axes to see that the graph is a parabola.

Using the discriminant, we see B^2-4AC = 0 so the graph is a parabola (rotated 75^@)