Given: The parametric equations below
[tex]\begin{gathered} x=t-4 \\ y=t^2+5t \end{gathered}[/tex]
To Determine: The rectangular equations representing the plane curve
Solution
Step 1: Make t the subject of the first parametric equation
[tex]\begin{gathered} x=t-4 \\ x+4=t \\ t=x+4 \end{gathered}[/tex]
Step 2: Substitute t in second parametric equation
[tex]\begin{gathered} y=t^2+5t \\ y=(x+4)^2+5(x+4) \\ y=(x+4)(x+4)+5x+20 \\ y=x^2+4x+4x+16+5x+20 \\ y=x^2+13x+36 \end{gathered}[/tex]
Hence, the equation of the plane curve is
y = x² + 13x + 36
