How do I solve this?

Question

7x^2 + 2(sqrt3)xy + 5y^2 = 8

Use a rotation of axes to eliminate the xy-term. (Write an equation in XY-coordinates. Use a rotation angle that satisfies 0 ≤ 𝜑 ≤ 𝜋/2.)

Answer 1

here's the solution to a similar problem with the XY term negative instead of positive

It's an ellipse with right and left ends (+/-1.27974, 0) centered at the origin

and top&bottom max and min = (0, +/-1.51421)

that gives about (x^2)/1.27974^2 + (y^2)/1.51421^2 = 1

or about (x^2)/1.6 + (y^2)/2.3= 1

or about 2.3x^2 +1.6y^2 = 1.9

cot2T = (A-C)/B = (7-5)/2sqr3 = 1/sqr3 = (sqr3)/3

tan2T = sqr3

2T = arctan(sqr3) = pi/3 = 60 degrees

T = pi/6 = 30 degrees

sin30 = 1/2, cos30 = sqr3/2

x = x'sin30 + y'cos30= x'/2 + y'sqr3/2

y = x'cos30 - y' sin30 = x'/2 - y'spr3/2

for the given problem with positive XY term, you get a similar ellipse centered at the origin but smaller,not as high or wide

Answer 2

Ax² + Bxy + Cy² + Dx + Ey + F = 0

cot (2θ) = (A - C) / B

Determine θ, sin θ, cosθ

x = x' cos θ - y' sin θ

y = x' sin θ + y' cos θ

Sustitute valus and simplify.