11 x1 t11 09 locus problems (2013)
-
Upload
nigel-simmons -
Category
Education
-
view
620 -
download
3
Transcript of 11 x1 t11 09 locus problems (2013)
![Page 1: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/1.jpg)
Locus Problems
![Page 2: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/2.jpg)
Locus Problems Eliminate the parameter (get rid of the p’s and q’s)
![Page 3: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/3.jpg)
Locus Problems We find the locus of a point.
Eliminate the parameter (get rid of the p’s and q’s)
![Page 4: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/4.jpg)
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)
![Page 5: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/5.jpg)
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)
![Page 6: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/6.jpg)
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)
![Page 7: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/7.jpg)
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)
![Page 8: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/8.jpg)
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)
![Page 9: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/9.jpg)
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)
![Page 10: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/10.jpg)
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)
![Page 11: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/11.jpg)
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)
![Page 12: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/12.jpg)
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)
![Page 13: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/13.jpg)
c) Type 3: not an obvious relationship between the values, use a
previously proven relationship between the parameters.
![Page 14: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/14.jpg)
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
![Page 15: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/15.jpg)
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
![Page 16: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/16.jpg)
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
![Page 17: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/17.jpg)
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
![Page 18: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/18.jpg)
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
![Page 19: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/19.jpg)
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
![Page 20: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/20.jpg)
(iii) Find the locus of R if PQ passes through (0,a)
![Page 21: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/21.jpg)
(iii) Find the locus of R if PQ passes through (0,a)
qpapqx
![Page 22: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/22.jpg)
(iii) Find the locus of R if PQ passes through (0,a)
qpapqx
a
xqp
qpax
![Page 23: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/23.jpg)
(iii) Find the locus of R if PQ passes through (0,a)
qpapqx
a
xqp
qpax
222 qpqpay
![Page 24: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/24.jpg)
(iii) Find the locus of R if PQ passes through (0,a)
qpapqx
a
xqp
qpax
222 qpqpay
22
pqqpay
![Page 25: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/25.jpg)
(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
![Page 26: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/26.jpg)
(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
![Page 27: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/27.jpg)
2004 Extension 1 HSC Q4b)
The two points and are on the parabola 2,2 apapP 2,2 aqaqQ ayx 42
![Page 28: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/28.jpg)
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 ,
![Page 29: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/29.jpg)
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 ,
![Page 30: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/30.jpg)
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
![Page 31: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/31.jpg)
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
![Page 32: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/32.jpg)
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
![Page 33: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/33.jpg)
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
![Page 34: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/34.jpg)
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
![Page 35: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/35.jpg)
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
![Page 36: 11 x1 t11 09 locus problems (2013)](https://reader033.fdocuments.net/reader033/viewer/2022042716/55abb2ea1a28abcf558b4868/html5/thumbnails/36.jpg)
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