Question

Eliminate the parameter and obtain the standard form of the rectangular equation.

Answer 1

Line through \[(x _{1},y _{1}) and (x _{2},y _{2}): x=x _{1}+t(_{x2}-_{x1}), y=y _{1}+t(y _{2}-y _{1})\]

Answer 2

Circle: \[x=h+r \cos \theta, y=k+r \sin \theta \]

Answer 3

For the first, you can eliminate \(t\) by solving for it in one of the equations, then substituting the resulting expression into the other equation. For example, take the first equation and you have
\[x=x_1+\color{red}t(x_2-x_1)~~\iff~~\color{red}t=\frac{x-x_1}{x_2-x_1}\]and go from there.

For the second, you can use a fundamental trigonometric identity. Recall that \(\cos^2\theta+\sin^2\theta=1\).
\[\begin{cases}x-h=r\cos\theta\\y-k=r\sin\theta\end{cases}~~\implies~~\begin{cases}(x-h)^2=\mathbin{?}\\(y-k)^2=\mathbin{?}\end{cases}~~\implies~~(x-h)^2+(y-k)^2=\mathbin{?}\]