Question

The parametric equations x = t − 4 and y = t 2 + 5t represent a plane curve. Which of the following rectangular equations represents the plane curve?

Answer 1

Given: The parametric equations below

`\begin{gathered} x=t-4 \\ y=t^2+5t \end{gathered}`

To Determine: The rectangular equations representing the plane curve

Solution

Step 1: Make t the subject of the first parametric equation

`\begin{gathered} x=t-4 \\ x+4=t \\ t=x+4 \end{gathered}`

Step 2: Substitute t in second parametric equation

`\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}`

Hence, the equation of the plane curve is

y = x² + 13x + 36