11X1 T12 09 locus problems

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  • Locus Problems

  • Locus Problems(get rid of the ps and qs)

  • Locus Problems(get rid of the ps and qs)

    We find the locus of a point.

  • Locus Problems(get rid of the ps and qs)

    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 ps and qs)

    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 ps and qs)

    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 ps and qs)

    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 ps and qs)

    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 ps and qs)

    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 ps and qs)

    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 ps and qs)

    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 ps and qs)

    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. 22Given that pq = 3

  • c) Type 3: not an obvious relationship between the values, use a previously proven relationship between the parameters. qpqp , e.g. 22Given that pq = 3 pqqp

    qpx

    2222

  • c) Type 3: not an obvious relationship between the values, use a previously proven relationship between the parameters. qpqp , e.g. 22Given that pq = 3 pqqp

    qpx

    2222

    62 yx

  • c) Type 3: not an obvious relationship between the values, use a previously proven relationship between the parameters. qpqp , e.g. 22Given that pq = 3 pqqp

    qpx

    2222

    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. 22Given that pq = 3 pqqp

    qpx

    2222

    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. 22Given that pq = 3 pqqp

    qpx

    2222

    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

    2122

    axay

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

    qpapqx

    axqp

    qpax

    222 qpqpay 22 pqqpay

    2122

    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 mm1

    020

    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 mm1

    020

    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 mm1

    020

    020 22

    aqaq

    apap

    pqaqap 222 44pq

  • 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 mm1

    020

    020 22

    aqaq

    apap

    pqaqap 222 44pq

    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 mm1

    020

    020 22

    aqaq

    apap

    pqaqap 222 44pq

    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 mm1

    020

    020 22

    aqaq

    apap

    pqaqap 222 44pq

    apqy ay 4

    Exercise 9J; oddsup to 23

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