Identify a parametric self-intersection

C

Total marks: 7

Solution

(i)

Parametric intercepts

A curve $C$ has parametric equations $x=t^2-1$ , $y=4t-t^3+15$ , where $t\in\mathbb R$ . Show that $C$ cuts the $x$ -axis at $(8,0)$ and the $y$ -axis at $(0,12)$ and $(0,18)$ .

Hint

$x$ -axis: put $y=0$ and find $t$ .
$y$ -axis: put $x=0$ , so $t=\pm1$ .

Answer · (i)

(8,0),(0,12),(0,18)

(ii)

Sketching parametric curves

Sketch $C$ , giving the equation of the line of symmetry.

Explore this idea

Hint

Replace $t$ by $-t$ : what happens to $x$ and to $y-15$ ?
A reflection that keeps $x$ and negates $y-15$ is a reflection in $y=15$ .

Q20 parametric crossing curveScroll across to see the full diagram.

Answer · (ii)

y=15

Explore the idea

A loop with a line of symmetry

The curve is drawn as t runs from −3.3 to 3.3.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

C
Line of symmetry y = 15
Axis intercepts
Self-intersection

Ideas to try

Follow the curve from the top arm into the loop. At which point does it cross itself?
Where would the curve cross the $x$ -axis a second time if it were not symmetric? (It does not.)

(iii)

Self-intersections

Find the exact coordinates of the point where $C$ crosses itself.

Hint

A self-intersection is a point reached by two different values of $t$ .
By symmetry it lies on $y=15$ . Solve $y=15$ for $t$ .
Reject the value that is not a crossing.

Answer · (iii)

(3,15)

Identify a parametric self-intersection

Solution

Parametric intercepts

$x$ -axis: $y=0$

$y$ -axis: $x=0$

Find the heights

Sketching parametric curves

Symmetry in $t\to-t$

Line of symmetry

Shape

A loop with a line of symmetry

Self-intersections

Use the symmetry

Candidates

Choose the crossing

xxx-axis: y=0y=0y=0

yyy-axis: x=0x=0x=0

Find the heights

Symmetry in t→−tt\to-tt→−t

Line of symmetry

Shape

Use the symmetry

Candidates

Choose the crossing

$x$ -axis: $y=0$

$y$ -axis: $x=0$

Symmetry in $t\to-t$