11X1 T12 09 locus problems (2011)

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  • Locus Problems

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

  • Locus Problems We find the locus of a point.

    Eliminate the parameter (get rid of the ps and qs)

  • 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 ps and qs)

  • 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 ps and qs)

  • 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 ps and qs)

  • 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 ps and qs)

  • 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 ps and qs)

  • 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 ps and qs)

  • 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 ps and qs)

  • 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 ps and qs)

  • 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 ps and qs)

  • 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