11X1 T11 07 chord of contact (2010)

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<ul><li><p>Chord of Contact </p></li><li><p>Chord of Contact </p><p>y </p><p>x </p><p>2 4x ay</p></li><li><p>Chord of Contact </p><p>y </p><p>x </p><p>2 4x ay</p><p>We know the coordinates </p><p>of an external point (T) </p></li><li><p>Chord of Contact </p><p>y </p><p>x </p><p>2 4x ay</p><p> 0 0,T x y</p><p>We know the coordinates </p><p>of an external point (T) </p></li><li><p>Chord of Contact </p><p>y </p><p>x </p><p>2 4x ay</p><p> 0 0,T x y</p><p>We know the coordinates </p><p>of an external point (T) </p><p>From this external point, </p><p>two tangents can be drawn </p><p>meeting the parabola at P </p><p>and Q. </p></li><li><p>Chord of Contact </p><p>y </p><p>x </p><p>2 4x ay</p><p>2(2 , )Q aq aq</p><p>2(2 , )P ap ap</p><p> 0 0,T x y</p><p>We know the coordinates </p><p>of an external point (T) </p><p>From this external point, </p><p>two tangents can be drawn </p><p>meeting the parabola at P </p><p>and Q. </p></li><li><p>Chord of Contact </p><p>y </p><p>x </p><p>2 4x ay</p><p>2(2 , )Q aq aq</p><p>2(2 , )P ap ap</p><p> 0 0,T x y</p><p>We know the coordinates </p><p>of an external point (T) </p><p>From this external point, </p><p>two tangents can be drawn </p><p>meeting the parabola at P </p><p>and Q. </p></li><li><p>Chord of Contact </p><p>y </p><p>x </p><p>2 4x ay</p><p>2(2 , )Q aq aq</p><p>2(2 , )P ap ap</p><p> 0 0,T x y</p><p>We know the coordinates </p><p>of an external point (T) </p><p>From this external point, </p><p>two tangents can be drawn </p><p>meeting the parabola at P </p><p>and Q. </p><p>The line joining these two </p><p>points is called the chord </p><p>of contact. </p></li><li><p>Chord of Contact </p><p>y </p><p>x </p><p>2 4x ay</p><p>2(2 , )Q aq aq</p><p>2(2 , )P ap ap</p><p> 0 0,T x y</p><p>We know the coordinates </p><p>of an external point (T) </p><p>From this external point, </p><p>two tangents can be drawn </p><p>meeting the parabola at P </p><p>and Q. </p><p>The line joining these two </p><p>points is called the chord </p><p>of contact. </p></li><li><p>(1) Parametric approach </p></li><li><p>(1) Parametric approach </p><p>1 Show that has equation 2 2PQ p q x y apq </p></li><li><p>(1) Parametric approach </p><p>1 Show that has equation 2 2PQ p q x y apq </p><p>2 </p><p>2 2</p><p>Show the two tangents have equations</p><p>0 and 0px y ap qx y aq </p></li><li><p>(1) Parametric approach </p><p>1 Show that has equation 2 2PQ p q x y apq </p><p>2 </p><p>2 2</p><p>Show the two tangents have equations</p><p>0 and 0px y ap qx y aq </p><p>3 Show that is the point ,T a p q apq</p></li><li><p>(1) Parametric approach </p><p>1 Show that has equation 2 2PQ p q x y apq </p><p>2 </p><p>2 2</p><p>Show the two tangents have equations</p><p>0 and 0px y ap qx y aq </p><p>3 Show that is the point ,T a p q apq</p><p>4 0 0But is ,T x y</p></li><li><p>(1) Parametric approach </p><p>1 Show that has equation 2 2PQ p q x y apq </p><p>2 </p><p>2 2</p><p>Show the two tangents have equations</p><p>0 and 0px y ap qx y aq </p><p>3 Show that is the point ,T a p q apq</p><p>4 0 0But is ,T x y </p><p>a</p><p>xqp</p><p>qpax</p><p>0</p><p>0</p></li><li><p>(1) Parametric approach </p><p>1 Show that has equation 2 2PQ p q x y apq </p><p>2 </p><p>2 2</p><p>Show the two tangents have equations</p><p>0 and 0px y ap qx y aq </p><p>3 Show that is the point ,T a p q apq</p><p>4 0 0But is ,T x y </p><p>a</p><p>xqp</p><p>qpax</p><p>0</p><p>0</p><p> apqy 0</p></li><li><p>(1) Parametric approach </p><p>1 Show that has equation 2 2PQ p q x y apq </p><p>2 </p><p>2 2</p><p>Show the two tangents have equations</p><p>0 and 0px y ap qx y aq </p><p>3 Show that is the point ,T a p q apq</p><p>4 0 0But is ,T x y </p><p>a</p><p>xqp</p><p>qpax</p><p>0</p><p>0</p><p>00 22 is yya</p><p>xxPQ </p><p>apqy 0</p></li><li><p>(1) Parametric approach </p><p>1 Show that has equation 2 2PQ p q x y apq </p><p>2 </p><p>2 2</p><p>Show the two tangents have equations</p><p>0 and 0px y ap qx y aq </p><p>3 Show that is the point ,T a p q apq</p><p>4 0 0But is ,T x y </p><p>a</p><p>xqp</p><p>qpax</p><p>0</p><p>0</p><p>00 22 is yya</p><p>xxPQ </p><p>Hence the chord of contact is yyaxx 00 2</p><p>apqy 0</p></li><li><p>(1) Parametric approach </p><p>1 Show that has equation 2 2PQ p q x y apq </p><p>2 </p><p>2 2</p><p>Show the two tangents have equations</p><p>0 and 0px y ap qx y aq </p><p>3 Show that is the point ,T a p q apq</p><p>4 0 0But is ,T x y </p><p>a</p><p>xqp</p><p>qpax</p><p>0</p><p>0</p><p>00 22 is yya</p><p>xxPQ </p><p>Hence the chord of contact is yyaxx 00 2</p><p>apqy 0</p><p>notice </p><p>similarity </p><p>to tangent </p></li><li><p>(2) Cartesian approach </p></li><li><p>(2) Cartesian approach </p><p>1 1 1Show that has equation 2PT xx a y y </p></li><li><p>(2) Cartesian approach </p><p>1 1 1Show that has equation 2PT xx a y y </p><p>T lies on PT 1010 2 yyaxx </p></li><li><p>(2) Cartesian approach </p><p>1 1 1Show that has equation 2PT xx a y y </p><p>T lies on PT 1010 2 yyaxx </p><p> yyaxxyxP 0011 2equation with line on the lies ,</p></li><li><p>(2) Cartesian approach </p><p>1 1 1Show that has equation 2PT xx a y y </p><p>T lies on PT 1010 2 yyaxx </p><p>2 2 2Show that has equation 2QT xx a y y </p><p> yyaxxyxP 0011 2equation with line on the lies ,</p></li><li><p>(2) Cartesian approach </p><p>1 1 1Show that has equation 2PT xx a y y </p><p>T lies on PT 1010 2 yyaxx </p><p>2 2 2Show that has equation 2QT xx a y y </p><p>T lies on QT 2020 2 yyaxx </p><p> yyaxxyxP 0011 2equation with line on the lies ,</p></li><li><p>(2) Cartesian approach </p><p>1 1 1Show that has equation 2PT xx a y y </p><p>T lies on PT 1010 2 yyaxx </p><p>2 2 2Show that has equation 2QT xx a y y </p><p>T lies on QT 2020 2 yyaxx </p><p> yyaxxyxP 0011 2equation with line on the lies ,</p><p> yyaxxyxQ 0022 2equation with line on the lies ,</p></li><li><p>(2) Cartesian approach </p><p>1 1 1Show that has equation 2PT xx a y y </p><p>T lies on PT 1010 2 yyaxx </p><p>2 2 2Show that has equation 2QT xx a y y </p><p>T lies on QT 2020 2 yyaxx </p><p> yyaxxyxP 0011 2equation with line on the lies ,</p><p> yyaxxyxQ 0022 2equation with line on the lies ,</p><p>Hence the chord of contact is yyaxx 00 2</p></li><li><p>(2) Cartesian approach </p><p>1 1 1Show that has equation 2PT xx a y y </p><p>T lies on PT 1010 2 yyaxx </p><p>2 2 2Show that has equation 2QT xx a y y </p><p>T lies on QT 2020 2 yyaxx </p><p> yyaxxyxP 0011 2equation with line on the lies ,</p><p> yyaxxyxQ 0022 2equation with line on the lies ,</p><p>Hence the chord of contact is yyaxx 00 2</p><p>Exercise 9H; 1c, 2d, 3, 6, 8, 10, 14 </p></li></ul>