2 Squareroot 3 x2 6xy Squareroot 3x 3y 0 Use the discrimi

2 Squareroot 3 x^2 - 6xy + Squareroot 3x + 3y =0 Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. parabola ellipse hyperbola. Use a rotation of axes to eliminate the xy-term. (Write an equation in XY-coordinates. Use a rotation angle that satisfies 0 lessthanorequalto phi lessthanorequalto pi/2.)

Solution

2 sqrt 3x^2 - 6xy + sqrt 3x + 3y = 0

A = 2 sqrt 3

B = -6

C = sqrt3

cot 2theta = A-C / B

Plugging the values we get

cot 2theta = (2 sqrt 3 - sqrt 3 ) / 6

cot 2 theta = sqrt 3 / 6

tan 2 theta = 6 / sqrt 3 = 6 sqrt 3 / 3 = 2 sqrt 3

2 theta = tan^-1 ( 2 sqrt 3 ) = 73.897

theta = 73.897 / 2 = 36.948 degrees

rotation angle = 36.948 degrees