Conversion of Parametric Equations to Cartesian Equations

Question 1

Find the Cartesian equation of \( x=t-2, \ y=t^2 \).

\( \begin{align} t &= x+2 \\ \therefore y &= (x+2)^2 \end{align} \)

Question 2

Find the Cartesian equation of \( x=6y, \ y=3t^2 \).

\( \begin{align} \displaystyle t &= \frac{x}{6} \\ y &= 3 \times \left(\frac{x}{6}\right)^2 = 3 \times \frac{x^2}{36} = \frac{x^2}{12} \\ \therefore y &= \frac{x^2}{12} \end{align} \)

Question 3

Find the Cartesian equation of \( \displaystyle x=\frac{1}{t}, \ y=2t \).

\( \begin{align} \displaystyle t &= \frac{1}{t} \\ y &= 2 \times \frac{1}{2} \\ \therefore y &= \frac{2}{x} \end{align} \)

Question 4

Find the Cartesian equation of \( x=2at, \ y=at^2 \).

\( \begin{align} \displaystyle t &= \frac{x}{2a} \\ y &= a \times \left(\frac{x}{2a}\right)^2 = a \times \frac{x^2}{4a^2} = \frac{x^2}{4a} \\ \therefore y &= \frac{x^2}{4a} \end{align} \)

Question 5

Find the Cartesian equation of \( \displaystyle x=t+\frac{1}{t}, \ y=t^2+\frac{1}{t^2} \).

\( \begin{align} \displaystyle x^2 &= \left(t+\frac{1}{t}\right)^2 = t^2+2+\frac{1}{t^2} = y+2 \\ x^2 &= y+2 \\ \therefore y &= x^2-2 \end{align} \)

Question 6

Find the Cartesian equation of \( x=2 \cos \theta, \ y = 2 \sin \theta \).

\( \require{AMSsymbols} \begin{align} x^2 &= 4 \cos^2 \theta \cdots (1) \\ y^2 &= 4 \sin^2 \theta \cdots (2) \\ x^2+y^2 &= 4 \cos^2 \theta + 4 \sin^2 \theta &\color{green}{(1)+(2)} \\ &= 4\left( \cos^2 \theta + \sin^2 \theta \right) = 4 \times 1 = 4 \\ \therefore x^2+y^2 &= 4 \end{align} \)

Question 7

Find the Cartesian equation of \( x=1+2 \tan \theta, \ y =3 \sec \theta-4 \).

\( \begin{align} \displaystyle 2 \tan \theta &= x-1 \\ \tan \theta &= \frac{x-1}{2} \cdots (1) \\ 3 \sec \theta &= y+4 \\ \sec \theta &= \frac{y+4}{3} \cdots (2) \\ (1)^2+1 &= (2)^2 &\color{green}{\tan^2 \theta + 1 = \sec^2 \theta} \\ \therefore \frac{(x-1)^2}{4}+1 &= \frac{y+4}{9} \end{align} \)

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