Question

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

Answer 1

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^@`)