I had a bit of trouble on this question where it describes to express the following equations in just x and y. "Eliminate $\theta$ from the following equations :" $$ x\sin\theta + y\cos\theta = a , y\sin\theta -x\cos\theta =b $$
I have done similar ones before but the inability to eliminate the x,y/ additional variable is making it confusing.
Thank you
We have $$x\sin \theta + y\cos \theta = a\tag{1}$$ $$-x\cos \theta + y\sin \theta = b\tag{2}$$ Squaring $(1)$ and $(2)$, we get, $$x^2\sin^2 \theta + 2xy\sin \theta\cos \theta + y^2\cos^2 \theta = a^2\tag{3}$$ $$x^2\cos^2\theta -2xy\sin \theta\cos \theta + y^2\sin^2 \theta = b^2\tag{4}$$ $$(3)+(4)\implies x^2(\sin^2\theta + \cos^2\theta) +y^2(\sin^2 \theta + \cos^2 \theta) = a^2+b^2$$ $$\boxed{x^2 + y^2 = a^2 + b^2}$$
Hope it helps.
Hint: Bring $\sin \theta$ and $\cos \theta$ in terms of $x$ and $y$. Then use the identity $\sin^2 \theta + \cos^2 \theta =1$.