Locus Problems

Locus Problems Eliminate the parameter (get rid of the p’s and q’s)

Locus Problems We find the locus of a point.

Eliminate the parameter (get rid of the p’s and q’s)

Locus Problems We find the locus of a point.

1 Find the coordinates of the point whose locus you are finding.

Eliminate the parameter (get rid of the p’s and q’s)

Locus Problems We find the locus of a point.

1

2

Find the coordinates of the point whose locus you are finding.

Look for the relationship between the x and y values

(i.e. get rid of the parameters)

Eliminate the parameter (get rid of the p’s and q’s)

Locus Problems We find the locus of a point.

1

2

Find the coordinates of the point whose locus you are finding.

Look for the relationship between the x and y values

(i.e. get rid of the parameters)

a) Type 1: already no parameter in the x or y value.

Eliminate the parameter (get rid of the p’s and q’s)

Locus Problems We find the locus of a point.

1

2

Find the coordinates of the point whose locus you are finding.

Look for the relationship between the x and y values

(i.e. get rid of the parameters)

a) Type 1: already no parameter in the x or y value.

e.g. (p + q,– 3)

Eliminate the parameter (get rid of the p’s and q’s)

Locus Problems We find the locus of a point.

1

2

Find the coordinates of the point whose locus you are finding.

Look for the relationship between the x and y values

(i.e. get rid of the parameters)

a) Type 1: already no parameter in the x or y value.

e.g. (p + q,– 3)

locus is y = – 3

Eliminate the parameter (get rid of the p’s and q’s)

Locus Problems We find the locus of a point.

1

2

Find the coordinates of the point whose locus you are finding.

Look for the relationship between the x and y values

(i.e. get rid of the parameters)

a) Type 1: already no parameter in the x or y value.

e.g. (p + q,– 3)

locus is y = – 3

b) Type 2: obvious relationship between the values, usually only one

parameter.

Eliminate the parameter (get rid of the p’s and q’s)

Locus Problems We find the locus of a point.

1

2

Find the coordinates of the point whose locus you are finding.

Look for the relationship between the x and y values

(i.e. get rid of the parameters)

a) Type 1: already no parameter in the x or y value.

e.g. (p + q,– 3)

locus is y = – 3

b) Type 2: obvious relationship between the values, usually only one

parameter.

1,6 e.g. 2 tt

Eliminate the parameter (get rid of the p’s and q’s)

Locus Problems We find the locus of a point.

1

2

Find the coordinates of the point whose locus you are finding.

Look for the relationship between the x and y values

(i.e. get rid of the parameters)

a) Type 1: already no parameter in the x or y value.

e.g. (p + q,– 3)

locus is y = – 3

b) Type 2: obvious relationship between the values, usually only one

parameter.

1,6 e.g. 2 tt

6

6

xt

tx

Eliminate the parameter (get rid of the p’s and q’s)

Locus Problems We find the locus of a point.

1

2

Find the coordinates of the point whose locus you are finding.

Look for the relationship between the x and y values

(i.e. get rid of the parameters)

a) Type 1: already no parameter in the x or y value.

e.g. (p + q,– 3)

locus is y = – 3

b) Type 2: obvious relationship between the values, usually only one

parameter.

1,6 e.g. 2 tt

6

6

xt

tx

136

1

2

2

xy

ty

Eliminate the parameter (get rid of the p’s and q’s)

c) Type 3: not an obvious relationship between the values, use a

previously proven relationship between the parameters.

c) Type 3: not an obvious relationship between the values, use a

previously proven relationship between the parameters.

qpqp , e.g. 22

Given that pq = 3

c) Type 3: not an obvious relationship between the values, use a

previously proven relationship between the parameters.

qpqp , e.g. 22

Given that pq = 3 pqqp

qpx

22

22

c) Type 3: not an obvious relationship between the values, use a

previously proven relationship between the parameters.

qpqp , e.g. 22

Given that pq = 3 pqqp

qpx

22

22

62 yx

c) Type 3: not an obvious relationship between the values, use a

previously proven relationship between the parameters.

qpqp , e.g. 22

Given that pq = 3 pqqp

qpx

22

22

62 yx

2005 Extension 1 HSC Q4c)

The points and lie on the parabola

The equation of the normal to the parabola at P is

and the equation of the normal at Q is given by

2,2 apapP 2,2 aqaqQ ayx 42 32 apappyx

32 aqaqqyx

c) Type 3: not an obvious relationship between the values, use a

previously proven relationship between the parameters.

qpqp , e.g. 22

Given that pq = 3 pqqp

qpx

22

22

62 yx

2005 Extension 1 HSC Q4c)

The points and lie on the parabola

The equation of the normal to the parabola at P is

and the equation of the normal at Q is given by

2,2 apapP 2,2 aqaqQ ayx 42 32 apappyx

32 aqaqqyx

(i) Show that the normals at P and Q intersect at the point R whose

coordinates are 2, 22 qpqpaqpapq

c) Type 3: not an obvious relationship between the values, use a

previously proven relationship between the parameters.

qpqp , e.g. 22

Given that pq = 3 pqqp

qpx

22

22

62 yx

2005 Extension 1 HSC Q4c)

The points and lie on the parabola

The equation of the normal to the parabola at P is

and the equation of the normal at Q is given by

2,2 apapP 2,2 aqaqQ ayx 42 32 apappyx

32 aqaqqyx

(i) Show that the normals at P and Q intersect at the point R whose

coordinates are 2, 22 qpqpaqpapq

(ii) The equation of the chord PQ is . If PQ passes

through (0,a), show that pq = – 1

apqxqpy 2

1

(iii) Find the locus of R if PQ passes through (0,a)

(iii) Find the locus of R if PQ passes through (0,a)

qpapqx

(iii) Find the locus of R if PQ passes through (0,a)

qpapqx

a

xqp

qpax

(iii) Find the locus of R if PQ passes through (0,a)

qpapqx

a

xqp

qpax

222 qpqpay

(iii) Find the locus of R if PQ passes through (0,a)

qpapqx

a

xqp

qpax

222 qpqpay

22

pqqpay

(iii) Find the locus of R if PQ passes through (0,a)

qpapqx

a

xqp

qpax

222 qpqpay

22

pqqpay

21

2

2

a

xay

(iii) Find the locus of R if PQ passes through (0,a)

qpapqx

a

xqp

qpax

222 qpqpay

22

pqqpay

21

2

2

a

xay

aa

xy 3

2

2004 Extension 1 HSC Q4b)

The two points and are on the parabola 2,2 apapP 2,2 aqaqQ ayx 42

2004 Extension 1 HSC Q4b)

The two points and are on the parabola 2,2 apapP 2,2 aqaqQ ayx 42

2attxy (i) The equation of the tangent to at an arbitrary point

on the parabola is

Show that the tangents at the points P and Q meet at R, where R is

the point

ayx 42 2,2 atat

apqqpa ,

2004 Extension 1 HSC Q4b)

The two points and are on the parabola 2,2 apapP 2,2 aqaqQ ayx 42

2attxy (i) The equation of the tangent to at an arbitrary point

on the parabola is

Show that the tangents at the points P and Q meet at R, where R is

the point

(ii) As P varies, the point Q is chosen so that is a right angle,

where O is the origin.

Find the locus of R.

POQ

ayx 42 2,2 atat

apqqpa ,

2004 Extension 1 HSC Q4b)

The two points and are on the parabola 2,2 apapP 2,2 aqaqQ ayx 42

2attxy (i) The equation of the tangent to at an arbitrary point

on the parabola is

Show that the tangents at the points P and Q meet at R, where R is

the point

(ii) As P varies, the point Q is chosen so that is a right angle,

where O is the origin.

Find the locus of R.

POQ

ayx 42 2,2 atat

apqqpa ,

1 OQOP mm

2004 Extension 1 HSC Q4b)

The two points and are on the parabola 2,2 apapP 2,2 aqaqQ ayx 42

2attxy (i) The equation of the tangent to at an arbitrary point

on the parabola is

Show that the tangents at the points P and Q meet at R, where R is

the point

(ii) As P varies, the point Q is chosen so that is a right angle,

where O is the origin.

Find the locus of R.

POQ

ayx 42 2,2 atat

apqqpa ,

1 OQOP mm

102

0

02

0 22

aq

aq

ap

ap

2004 Extension 1 HSC Q4b)

The two points and are on the parabola 2,2 apapP 2,2 aqaqQ ayx 42

2attxy (i) The equation of the tangent to at an arbitrary point

on the parabola is

Show that the tangents at the points P and Q meet at R, where R is

the point

(ii) As P varies, the point Q is chosen so that is a right angle,

where O is the origin.

Find the locus of R.

POQ

ayx 42 2,2 atat

apqqpa ,

1 OQOP mm

102

0

02

0 22

aq

aq

ap

ap

pqaqap 222 4

2004 Extension 1 HSC Q4b)

The two points and are on the parabola 2,2 apapP 2,2 aqaqQ ayx 42

2attxy (i) The equation of the tangent to at an arbitrary point

on the parabola is

Show that the tangents at the points P and Q meet at R, where R is

the point

(ii) As P varies, the point Q is chosen so that is a right angle,

where O is the origin.

Find the locus of R.

POQ

ayx 42 2,2 atat

apqqpa ,

1 OQOP mm

102

0

02

0 22

aq

aq

ap

ap

pqaqap 222 4

4pq

2004 Extension 1 HSC Q4b)

The two points and are on the parabola 2,2 apapP 2,2 aqaqQ ayx 42

2attxy (i) The equation of the tangent to at an arbitrary point

on the parabola is

Show that the tangents at the points P and Q meet at R, where R is

the point

(ii) As P varies, the point Q is chosen so that is a right angle,

where O is the origin.

Find the locus of R.

POQ

ayx 42 2,2 atat

apqqpa ,

1 OQOP mm

102

0

02

0 22

aq

aq

ap

ap

pqaqap 222 4

4pq

apqy

2004 Extension 1 HSC Q4b)

The two points and are on the parabola 2,2 apapP 2,2 aqaqQ ayx 42

2attxy (i) The equation of the tangent to at an arbitrary point

on the parabola is

Show that the tangents at the points P and Q meet at R, where R is

the point

(ii) As P varies, the point Q is chosen so that is a right angle,

where O is the origin.

Find the locus of R.

POQ

ayx 42 2,2 atat

apqqpa ,

1 OQOP mm

102

0

02

0 22

aq

aq

ap

ap

pqaqap 222 4

4pq

apqy

ay 4

2004 Extension 1 HSC Q4b)

The two points and are on the parabola 2,2 apapP 2,2 aqaqQ ayx 42

2attxy (i) The equation of the tangent to at an arbitrary point

on the parabola is

Show that the tangents at the points P and Q meet at R, where R is

the point

(ii) As P varies, the point Q is chosen so that is a right angle,

where O is the origin.

Find the locus of R.

POQ

ayx 42 2,2 atat

apqqpa ,

1 OQOP mm

102

0

02

0 22

aq

aq

ap

ap

pqaqap 222 4

4pq

apqy

ay 4Exercise 9J; odds

up to 23