Download - 11 x1 t11 09 locus problems (2013)
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
