Download - 11X1 T12 09 locus problems
Locus Problems
Locus Problems(get rid of the p’s and q’s)
Locus Problems(get rid of the p’s and q’s)
We find the locus of a point.
Locus Problems(get rid of the p’s and q’s)
We find the locus of a point.
1 Find the coordinates of the point whose locus you are finding.
Locus Problems(get rid of the p’s and q’s)
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)
Locus Problems(get rid of the p’s and q’s)
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.
Locus Problems(get rid of the p’s and q’s)
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 Problems(get rid of the p’s and q’s)
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
Locus Problems(get rid of the p’s and q’s)
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.
Locus Problems(get rid of the p’s and q’s)
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
Locus Problems(get rid of the p’s and q’s)
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
6xt
tx
Locus Problems(get rid of the p’s and q’s)
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
6xt
tx
136
12
2
xy
ty
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 parabolaThe 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 parabolaThe 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 parabolaThe 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 21
(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
axqp
qpax
(iii) Find the locus of R if PQ passes through (0,a)
qpapqx
axqp
qpax
222 qpqpay
(iii) Find the locus of R if PQ passes through (0,a)
qpapqx
axqp
qpax
222 qpqpay
22 pqqpay
(iii) Find the locus of R if PQ passes through (0,a)
qpapqx
axqp
qpax
222 qpqpay
22 pqqpay
212
2
axay
(iii) Find the locus of R if PQ passes through (0,a)
qpapqx
axqp
qpax
222 qpqpay
22 pqqpay
212
2
axay
aaxy 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
1020
020 22
aqaq
apap
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
1020
020 22
aqaq
apap
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
1020
020 22
aqaq
apap
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
1020
020 22
aqaq
apap
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
1020
020 22
aqaq
apap
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
1020
020 22
aqaq
apap
pqaqap 222 4
4pq
apqy
ay 4Exercise 9J; odds
up to 23
