I need help with a pre-calculus qn?Pretty please!?

Question

Determine the appropriate rotation formulas to use so that the new equation contains no xy-term. x² + 4xy + 4y² + 5√5y +5= 0

Answer 1

Let u, v be the rotated x, y, rotated counter-clockwise by an angle t.

Then x = u*cos(t) + v*sin(t), and y = -u*sin(t) + v*cos(t).

Plug these into the equation (abbreviating c=cos(t), s=sin(t))

(uc+vs)^2 + 4(uc+vs)(vc-us) + 4(vc-us)^2 + (5*sqrt(5))(vc-us) + 5
= u^2*c^2 + 2*uvcs + v^2*s^2 + 4*uvc^2 + 4*v^2*cs - 4*u^2*cs - 4*uv*s^2 + 4*v^2*c^2 - 8*uvcs + 4*u^2*s^2 + 5*sqrt(5)(vc-us) + 5
= u^2(c^2-4cs+4s^2) + uv(2cs + 4c^2 - 4s^2 - 8cs) + v^2(s^2 + 4cs + 4c^2) + 5*sqrt(5)(vc-us) + 5

so to eliminate the xy term, we must have 2cs + 4c^2 - 4s^2 - 8cs = 0, but 2cs = sin(2t) and c^2 - s^2 = cos(2t), so we have
4cos(2t) = 3sin(2t), or tan(2t) = 4/3, so t = arctan(4/3)/2 = 0.4636 radians.

The resulting equation is 5v^2 - 5u + 10v + 5 = 0.

Answer 2

try the following:
x = u cos(t) + v sin(t), and y = -u sin(t) + v cos(t)
then
x^2 + 4xy +4y^2 +5sqrt(5) y +5 =0 becomes:
u^2cos^2 t + 2uvsintcost+ v^2 sin^2 t
+ 4( u cos(t) + v sin(t))(-u sin(t) + v cos(t)) +
4(u^2sin^2t - 2uvsintcost + v^2cos^2 t) +
5sqrt(5) (-usint + vcost ) + 5=0
you factor things together,
the coefficient of uv should be =0.
2sintcost +4 cos^2 t -sin^2t -8 sintcost =0
-6 sint cost + 4 cos^2 t - (1-cos^2 t) =0
-6 sint cost + 5 cos^2 t -1 =0
s