1. Geometry
Parametric equations

2. Starter: challenge

3. Parametric equations: Cartesian equations
KUS objectives
BAT convert between parametric and Cartesian equations of a function
Starter: previous page
Geogebra: parametric eqns

4. WB5 Parametric eqns – cartesian eqn example I
Find the Cartesian equations of the following curves
a) 𝑥=𝑡−1 𝑦= 2𝑡 2 +3𝑡 𝑡∈ℛ
b) 𝑥= 𝑡 2 𝑦= 2𝑡 2 +3𝑡 𝑡≥0

5. WB6 find the Cartesian equations of these (eliminate the parameters)
answers
𝑥=4𝑡 , 𝑦=3−𝑡
𝑥=𝑡−1 ,𝑦= 𝑡 2 +1
𝑥=2 𝑡 2 , 𝑦=4(𝑡−1)
𝑥=𝑡+2 , 𝑦= 1 𝑡
𝑥= 𝑡 2 −1 , 𝑦= 𝑡 2 +1
𝑥= 𝑡 2 −1 , 𝑦= 𝑡 4 +1 i)
ii)
iii)
iv) v)
vi)

6. WB 7 Draw the curve given by the Parametric Equations: x= 1 𝑡+1 𝑦= 𝑡 2 for −3≤𝑡≤3-3 -2 -1 1 2 3
x = 1/(t+1)
-1/2 -1  1
1/3
y = t2 9 4 1 1 4 9
Hmmmm… is this enough to sketch this graph?
equation of the curve?
𝑥= 1 𝑡+1
𝑦= 𝑡 2
𝑦= 1 𝑥 −1 2
𝑥(𝑡+1)=1
𝑡+1= 1 𝑥
𝑦= 1 𝑥 − 𝑥 𝑥 2
𝑡= 1 𝑥 −1
𝑦= 1−𝑥 𝑥 2
𝑦= (1−𝑥) 𝑥 2 2

7. WB8 Trigonometric Parametric eqns – cartesian eqn
Find the Cartesian equation of the following curve
𝑥= 𝑠𝑖𝑛 𝑡 𝑦= cos 𝑡 −𝜋≤𝑡≤ 𝜋

8. A curve has Parametric equations: 𝑥=𝑠𝑖𝑛𝑡+2 𝑦=𝑐𝑜𝑠𝑡−3
WB 9
A curve has Parametric equations: 𝑥=𝑠𝑖𝑛𝑡 𝑦=𝑐𝑜𝑠𝑡−3
a) Find the Cartesian equation of the curve
b) Sketch the curve
A Cartesian equation is just an equation of a line where the variables used are x and y only
𝑥=𝑠𝑖𝑛𝑡+2
𝑦=𝑐𝑜𝑠𝑡−3
𝑥−2=𝑠𝑖𝑛𝑡
𝑦+3=𝑐𝑜𝑠𝑡
How can we link sin t and cos t
in an equation?
𝑠𝑖 𝑛 2 𝑡+𝑐𝑜 𝑠 2 𝑡≡1
(𝑥−2 ) 2 +(𝑦+3 ) 2 = 1
The equation is that of a circle
 Think about where the centre will be, and its radius 5
(𝑥−2 ) 2 +(𝑦+3 ) 2 =1
Centre = (2, -3)
Radius = 1 -5 5 -5

9. Another way of writing this (by squaring the whole of each side)
WB 10
A curve has Parametric equations: 𝑥=𝑠𝑖𝑛𝑡 𝑦= sin 2𝑡
a) Find the Cartesian equation of the curve
b) Sketch the curve
𝑥=𝑠𝑖𝑛𝑡
𝑦=𝑠𝑖𝑛2𝑡
double angle formula
𝑥 2 =𝑠𝑖 𝑛 2 𝑡
𝑦=2𝑠𝑖𝑛𝑡𝑐𝑜𝑠𝑡
Replace sint with x
𝑦=2𝑥𝑐𝑜𝑠𝑡
𝑠𝑖 𝑛 2 𝑡+𝑐𝑜 𝑠 2 𝑡≡1
b) Geogebra: parametric eqns
𝑐𝑜 𝑠 2 𝑡=1−𝑠𝑖 𝑛 2 𝑡
Replace sin2t with x2
𝑐𝑜 𝑠 2 𝑡=1− 𝑥 2
𝑐𝑜𝑠𝑡= 1− 𝑥 2
𝑦=2𝑥𝑐𝑜𝑠𝑡
We can now replace cos t
𝑦=2𝑥 1− 𝑥 2
Another way of writing this (by squaring the whole of each side)
𝑦 2 =4 𝑥 2 (1− 𝑥 2 )

10. WB11 find the Cartesian equations of these (eliminate the parameters)
answers
𝑥=3 sin 𝑡 , 𝑦=2 cos 𝑡
𝑥= sec 𝑡 ,𝑦=5 tan 𝑡
𝑥=1+ cos 𝑡 ,
𝑦=1−2 sin 𝑡
𝑖𝑣) 𝑥= cos 𝑡 + sin 𝑡 ,
𝑦=2 cos 𝑡 + sin 𝑡
𝑣) 𝑥= cos 𝑡+ 𝜋 4 ,
𝑦= 2 sin 𝑡
𝑣𝑖) 𝑥=2 cos 𝑡 −1 ,
𝑦=3+2 sin 𝑡 i)
ii)
iii)
iv) v)
vi)

11. Parametric equations –Summary
Parametric equations are written as:
A Cartesian equation would be
There are three main types of question in the exam
Sketch a graph from parametric equations
Eliminate (t) to find the Cartesian equation
Differentiating to find gradients, tangents and normals
Make a table of values – for t, x and y and plot points (x, y)
‘Zoom in’ on any interesting points – work out more values
e.g. to check asymptotes are correct
Write one thing you have learned
Write one thing you need to improve