Finding cartesian equation for trigonometric parametric forms

Question

I'm trying to find the cartesian equation for these parameteric forms:

$$ x = sin\theta + 2 cos \theta \\ y = 2 sin\theta + cos\theta $$

I tried:

$$\begin{align} x^2 & = sin^2\theta + 4cos^2\theta \\ & = 1 - cos^2\theta + 4cos^2\theta \\ & = 1 + 3cos^2\theta \\ \\ y^2 & = 4sin^2\theta+cos^2\theta\\ & = 4(1 - cos^2\theta) + cos^2\theta \\ & = 4 - 3cos^2\theta \\ \\ \therefore & \space4 - y^2 = x^2 - 1\\ \space & x^2 + y^2 = 5 \end{align}$$

Which differs from the given answer of $5x^2 + 5y^2 - 8xy = 9$. Where am I going wrong?

Answer 1

Note that $$x^2\not=\sin^2\theta+4\cos^2\theta$$ and that $$x^2=\sin^2\theta+4\cos^2\theta+4\sin\theta\cos\theta$$ because $$(a+b)^2=a^2+b^2+\color{red}{2ab}.$$

Since we can get $$\sin\theta=\frac{-x+2y}{3},\ \ \ \cos\theta=\frac{2x-y}{3},$$ then use $$\sin^2\theta+\cos^2\theta=1.$$

Answer 2

In general, $(a+b)^2\ne a^2+b^2$ unless $ab=0$

Solve for $\sin\theta,\cos\theta$ in terms of $x,y$

Then use $\sin^2\theta+\cos^2\theta=1$ to eliminate $\theta$