11x1 t11 07 chord of contact (2010)

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  • Chord of Contact

  • Chord of Contact

    y

    x

    2 4x ay

  • Chord of Contact

    y

    x

    2 4x ay

    We know the coordinates

    of an external point (T)

  • Chord of Contact

    y

    x

    2 4x ay

    0 0,T x y

    We know the coordinates

    of an external point (T)

  • Chord of Contact

    y

    x

    2 4x ay

    0 0,T x y

    We know the coordinates

    of an external point (T)

    From this external point,

    two tangents can be drawn

    meeting the parabola at P

    and Q.

  • Chord of Contact

    y

    x

    2 4x ay

    2(2 , )Q aq aq

    2(2 , )P ap ap

    0 0,T x y

    We know the coordinates

    of an external point (T)

    From this external point,

    two tangents can be drawn

    meeting the parabola at P

    and Q.

  • Chord of Contact

    y

    x

    2 4x ay

    2(2 , )Q aq aq

    2(2 , )P ap ap

    0 0,T x y

    We know the coordinates

    of an external point (T)

    From this external point,

    two tangents can be drawn

    meeting the parabola at P

    and Q.

  • Chord of Contact

    y

    x

    2 4x ay

    2(2 , )Q aq aq

    2(2 , )P ap ap

    0 0,T x y

    We know the coordinates

    of an external point (T)

    From this external point,

    two tangents can be drawn

    meeting the parabola at P

    and Q.

    The line joining these two

    points is called the chord

    of contact.

  • Chord of Contact

    y

    x

    2 4x ay

    2(2 , )Q aq aq

    2(2 , )P ap ap

    0 0,T x y

    We know the coordinates

    of an external point (T)

    From this external point,

    two tangents can be drawn

    meeting the parabola at P

    and Q.

    The line joining these two

    points is called the chord

    of contact.

  • (1) Parametric approach

  • (1) Parametric approach

    1 Show that has equation 2 2PQ p q x y apq

  • (1) Parametric approach

    1 Show that has equation 2 2PQ p q x y apq

    2

    2 2

    Show the two tangents have equations

    0 and 0px y ap qx y aq

  • (1) Parametric approach

    1 Show that has equation 2 2PQ p q x y apq

    2

    2 2

    Show the two tangents have equations

    0 and 0px y ap qx y aq

    3 Show that is the point ,T a p q apq

  • (1) Parametric approach

    1 Show that has equation 2 2PQ p q x y apq

    2

    2 2

    Show the two tangents have equations

    0 and 0px y ap qx y aq

    3 Show that is the point ,T a p q apq

    4 0 0But is ,T x y

  • (1) Parametric approach

    1 Show that has equation 2 2PQ p q x y apq

    2

    2 2

    Show the two tangents have equations

    0 and 0px y ap qx y aq

    3 Show that is the point ,T a p q apq

    4 0 0But is ,T x y

    a

    xqp

    qpax

    0

    0

  • (1) Parametric approach

    1 Show that has equation 2 2PQ p q x y apq

    2

    2 2

    Show the two tangents have equations

    0 and 0px y ap qx y aq

    3 Show that is the point ,T a p q apq

    4 0 0But is ,T x y

    a

    xqp

    qpax

    0

    0

    apqy 0

  • (1) Parametric approach

    1 Show that has equation 2 2PQ p q x y apq

    2

    2 2

    Show the two tangents have equations

    0 and 0px y ap qx y aq

    3 Show that is the point ,T a p q apq

    4 0 0But is ,T x y

    a

    xqp

    qpax

    0

    0

    00 22 is yya

    xxPQ

    apqy 0

  • (1) Parametric approach

    1 Show that has equation 2 2PQ p q x y apq

    2

    2 2

    Show the two tangents have equations

    0 and 0px y ap qx y aq

    3 Show that is the point ,T a p q apq

    4 0 0But is ,T x y

    a

    xqp

    qpax

    0

    0

    00 22 is yya

    xxPQ

    Hence the chord of contact is yyaxx 00 2

    apqy 0

  • (1) Parametric approach

    1 Show that has equation 2 2PQ p q x y apq

    2

    2 2

    Show the two tangents have equations

    0 and 0px y ap qx y aq

    3 Show that is the point ,T a p q apq

    4 0 0But is ,T x y

    a

    xqp

    qpax

    0

    0

    00 22 is yya

    xxPQ

    Hence the chord of contact is yyaxx 00 2

    apqy 0

    notice

    similarity

    to tangent

  • (2) Cartesian approach

  • (2) Cartesian approach

    1 1 1Show that has equation 2PT xx a y y

  • (2) Cartesian approach

    1 1 1Show that has equation 2PT xx a y y

    T lies on PT 1010 2 yyaxx

  • (2) Cartesian approach

    1 1 1Show that has equation 2PT xx a y y

    T lies on PT 1010 2 yyaxx

    yyaxxyxP 0011 2equation with line on the lies ,

  • (2) Cartesian approach

    1 1 1Show that has equation 2PT xx a y y

    T lies on PT 1010 2 yyaxx

    2 2 2Show that has equation 2QT xx a y y

    yyaxxyxP 0011 2equation with line on the lies ,

  • (2) Cartesian approach

    1 1 1Show that has equation 2PT xx a y y

    T lies on PT 1010 2 yyaxx

    2 2 2Show that has equation 2QT xx a y y

    T lies on QT 2020 2 yyaxx

    yyaxxyxP 0011 2equation with line on the lies ,

  • (2) Cartesian approach

    1 1 1Show that has equation 2PT xx a y y

    T lies on PT 1010 2 yyaxx

    2 2 2Show that has equation 2QT xx a y y

    T lies on QT 2020 2 yyaxx

    yyaxxyxP 0011 2equation with line on the lies ,

    yyaxxyxQ 0022 2equation with line on the lies ,

  • (2) Cartesian approach

    1 1 1Show that has equation 2PT xx a y y

    T lies on PT 1010 2 yyaxx

    2 2 2Show that has equation 2QT xx a y y

    T lies on QT 2020 2 yyaxx

    yyaxxyxP 0011 2equation with line on the lies ,

    yyaxxyxQ 0022 2equation with line on the lies ,

    Hence the chord of contact is yyaxx 00 2

  • (2) Cartesian approach

    1 1 1Show that has equation 2PT xx a y y

    T lies on PT 1010 2 yyaxx

    2 2 2Show that has equation 2QT xx a y y

    T lies on QT 2020 2 yyaxx

    yyaxxyxP 0011 2equation with line on the lies ,

    yyaxxyxQ 0022 2equation with line on the lies ,

    Hence the chord of contact is yyaxx 00 2

    Exercise 9H; 1c, 2d, 3, 6, 8, 10, 14