Question

How do you convert the parametric equations into a Cartesian equation by eliminating the parameter $r$ : $x=r^2+r,y=r^2-r$ ?

Answer 1

Convert the parametric equations into a Cartesian equation:

Given equations are,

$$\begin{gathered} x=r^2+r...\left(1\right) \\ y=r^2-r...\left(2\right) \end{gathered}$$

Add equations $\left(1\right)\mathrm{and}\left(2\right)$:

$$\begin{gathered} \therefore x+y=2r^2 \\ \Rightarrow r^2=\frac{x+y}{2}...\left(3\right) \end{gathered}$$

Multiply equations $\left(1\right)\mathrm{and}\left(2\right)$:

$$\begin{gathered} \therefore xy=(r^2+r)(r^2-r) \\ \Rightarrow =r^4-r^2 \end{gathered}$$

Substitute the value of $r^2$ in above equation:

$$\begin{gathered} \therefore xy={\left(\frac{x+y}{2}\right)}^2-\left(\frac{x+y}{2}\right) \\ \Rightarrow xy=\frac{x^2+y^2+2xy}{4}-\frac{x+y}{2} \\ \Rightarrow 4xy=x^2+y^2+2xy-2x-2y \\ \Rightarrow x^2+y^2-2x-2y-2xy=0 \end{gathered}$$

Hence, the required cartesian equation is $x^2+y^2-2x-2y-2xy=0$