Parametrics Notes

7/21/2019 Parametrics Notes

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ParaMetRicEquAtioNs

Extension ONEExtension ONEExtension ONEExtension ONEPreliminary CoursePreliminary CoursePreliminary CoursePreliminary Course

Name _______________


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CAPACITY MATRIX EXTENSION 1TOPIC: Further Parametrics (Ext)

Your say!

What was the most important thing you learned?_________________________

What was something new you learnt? _________________________________

What part(s) of this topic will you need to work on? _____________________

CONTENT CAPACITY BREAKDOWN!DONE

IT!!!!

GOT

IT!!!!!

O

1. Review of the Parabola as a Locus Notes2. Deriving equations of chords, tangents

and normalsNotes

3. Applications of Parametric equations Ex 12.1 (Terry Lee)4. Properties of the parabola Ex 9.2 Q2, 55. Locus problems Ex 12.2 (Terry Lee)

Past HSC questions (Notes)


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The Parabola as a Locus!(A Review)

Put the following phrases into the correct order to complete a true statement for the

Parabola as a Locus.

of a point that is and the fixed line and a fixed lineThe fixed point The locus is called the Focus

is always a Parabola. equidistant from a fixed point is called the Directrix.

SUMMARY!

A parabola is equidistant from a fixed point and a fixed line.

TThe fixed line is called the _____________.

TThe fixed point is called the ______________.

TThe turning point of the parabola is called the ________________.

TThe axis of symmetry of the parabola is called its _______________.

TThe distance between the vertex and the focus is called the _____________________.

TAn interval joining any two points on the parabola is called a _______________.

TA chord that passes through the focus is called a _______________________.

TThe focal chord that is perpendicular to the axis is called the _____________________

TA ________________ is a straight line that touches the parabola at a single point.


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The locus of point P(x, y)moving so that it isequidistant from the point (0, a)and line y = -aisa parabola with equation x2= 4ay

TFocus at ________TDirectrix with equation ________TVertex at _______TAxis with equation ________TFocal length the distance from the vertex to

the focus with length _________

TLatus rectum that is a horizontal focal chordwith length ______

The locus of point P(x, y)moving so that it isequidistant from the point (0, -a)and line y = aisa parabola with equation x2= -4ay


the focus with length _________TLatus rectum that is a horizontal focal chord

with length ______

The locus of point P(x, y)moving so that it isequidistant from the point (a, 0)and line x = -aisa parabola with equation y2= 4ax





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The locus of point P(x, y)moving so that it isequidistant from the point (-a, 0)and line x = aisa parabola with equation y2= -4ax




The parabola with vertex (h, k)and axis parallelto the y-axis has equation (x h)2= 4a(y k)




The parabola with vertex (h, k)and axis parallelto the x-axis has equation (y k)2= 4a(x h)





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CHORDS, TANGENTS AND NORMALSCHORDS, TANGENTS AND NORMALSCHORDS, TANGENTS AND NORMALSCHORDS, TANGENTS AND NORMALS

e.g., Find the equation of the chord joining points where t=3 and t= -2 on the parabola x =

2at, y = at2

e.g., Find the equation of the tangent to the parabola x2= 8yat the point (4t, 2t2)


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PARAMETRIC FORMULAE

Rehash of previous work:

PROOF

PROOF

If P(2ap, ap) and Q(2aq, aq2) are any twopoints on the parabola x2= 4ay, then the chordPQ has gradient

2

qp +

and equation

( ) 02

1=++ apqxqpy

The parabola x2= 4aycan be

written asx = 2aty = at2

If PQ is a focal chord, then pq= -1

Learn toderive some

of theseequationsrather than


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The tangent to the parabola x2= 4ayat the point P(2ap, ap) has gradientpandequation given by

y px + ap2= 0.

The tangents to the parabola x2= 4ayat points P(2ap, ap) and Q(2aq,aq2) intersect at the point

[a(p + q), apq]

The normal to the curve x2= 4ayat point P(2ap, ap2) has gradient -p

1 and

equation given byx +py = ap3+ 2ap


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The equations of the tangent, normal and chord can also be derived from points in

Cartesian form rather than Parametric form.

The normals to the parabola x2= 4ayat P(2ap, ap2) and Q(2ap, aq2) intersectat

[-apq( p +q ), a( p2+pq + q2+2]

If point A(x1, y1) lies on the parabola ayx 42= , then the equation of the

tangent at A is given byxx1= 2a( y + y1)

If point A(x1, y1) lies on the parabola

= , then the equation of the

normal at A is given by

( )

=


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The equation of the chord of contact XY of tangents drawn from point

(x1, y1) to the parabola ayx 42= is given by

xx1= 2a(y + y1)


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e.g., Tangents are drawn from the point

2

1,

2

1to the points P and Q on the parabola

= . Find the equation of the chord of contact PQ and the co-ordinates of P and Q.


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Eg Find the locus of the midpoints of the chords in the parabola = that pass

through (0, 2).


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PROPERTIES OF A PARABOLAPROVE:

The tangents at the end of a focal chord INTERSECT

AT RIGHT ANGLES on the DIRECTRIX.

PROVE:

The tangent at point P on a parabola is equallyinclined to the axis of the parabola and the focal

chord through P.


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PRACTICE QUESTIONS (APPLICATION)


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SOLUTIONS


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PAST HSC QUESTIONS2009 HSC Q2


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EXT 1 HSC 2008 Q5


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EXT 1 HSC 2007 Q5


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EXT 1 HSC 2006 Q2


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PAST HSC SOLUTIONS2009 HSC Q2

EXT 1 HSC 2008 Q4


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EXT 1 HSC 2007 Q5

EXT 1 HSC 2006 Q2

Documents

Parametrics Notes