11 x1 t11 09 locus problems (2013)

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<ul><li><p>Locus Problems </p></li><li><p>Locus Problems Eliminate the parameter (get rid of the ps and qs) </p></li><li><p>Locus Problems We find the locus of a point. </p><p>Eliminate the parameter (get rid of the ps and qs) </p></li><li><p>Locus Problems We find the locus of a point. </p><p>1 Find the coordinates of the point whose locus you are finding. </p><p>Eliminate the parameter (get rid of the ps and qs) </p></li><li><p>Locus Problems We find the locus of a point. </p><p>1 </p><p>2 </p><p>Find the coordinates of the point whose locus you are finding. </p><p>Look for the relationship between the x and y values </p><p>(i.e. get rid of the parameters) </p><p>Eliminate the parameter (get rid of the ps and qs) </p></li><li><p>Locus Problems We find the locus of a point. </p><p>1 </p><p>2 </p><p>Find the coordinates of the point whose locus you are finding. </p><p>Look for the relationship between the x and y values </p><p>(i.e. get rid of the parameters) </p><p>a) Type 1: already no parameter in the x or y value. </p><p>Eliminate the parameter (get rid of the ps and qs) </p></li><li><p>Locus Problems We find the locus of a point. </p><p>1 </p><p>2 </p><p>Find the coordinates of the point whose locus you are finding. </p><p>Look for the relationship between the x and y values </p><p>(i.e. get rid of the parameters) </p><p>a) Type 1: already no parameter in the x or y value. </p><p>e.g. (p + q, 3) </p><p>Eliminate the parameter (get rid of the ps and qs) </p></li><li><p>Locus Problems We find the locus of a point. </p><p>1 </p><p>2 </p><p>Find the coordinates of the point whose locus you are finding. </p><p>Look for the relationship between the x and y values </p><p>(i.e. get rid of the parameters) </p><p>a) Type 1: already no parameter in the x or y value. </p><p>e.g. (p + q, 3) </p><p>locus is y = 3 </p><p>Eliminate the parameter (get rid of the ps and qs) </p></li><li><p>Locus Problems We find the locus of a point. </p><p>1 </p><p>2 </p><p>Find the coordinates of the point whose locus you are finding. </p><p>Look for the relationship between the x and y values </p><p>(i.e. get rid of the parameters) </p><p>a) Type 1: already no parameter in the x or y value. </p><p>e.g. (p + q, 3) </p><p>locus is y = 3 </p><p>b) Type 2: obvious relationship between the values, usually only one </p><p>parameter. </p><p>Eliminate the parameter (get rid of the ps and qs) </p></li><li><p>Locus Problems We find the locus of a point. </p><p>1 </p><p>2 </p><p>Find the coordinates of the point whose locus you are finding. </p><p>Look for the relationship between the x and y values </p><p>(i.e. get rid of the parameters) </p><p>a) Type 1: already no parameter in the x or y value. </p><p>e.g. (p + q, 3) </p><p>locus is y = 3 </p><p>b) Type 2: obvious relationship between the values, usually only one </p><p>parameter. </p><p> 1,6 e.g. 2 tt</p><p>Eliminate the parameter (get rid of the ps and qs) </p></li><li><p>Locus Problems We find the locus of a point. </p><p>1 </p><p>2 </p><p>Find the coordinates of the point whose locus you are finding. </p><p>Look for the relationship between the x and y values </p><p>(i.e. get rid of the parameters) </p><p>a) Type 1: already no parameter in the x or y value. </p><p>e.g. (p + q, 3) </p><p>locus is y = 3 </p><p>b) Type 2: obvious relationship between the values, usually only one </p><p>parameter. </p><p> 1,6 e.g. 2 tt</p><p>6</p><p>6</p><p>xt</p><p>tx</p><p>Eliminate the parameter (get rid of the ps and qs) </p></li><li><p>Locus Problems We find the locus of a point. </p><p>1 </p><p>2 </p><p>Find the coordinates of the point whose locus you are finding. </p><p>Look for the relationship between the x and y values </p><p>(i.e. get rid of the parameters) </p><p>a) Type 1: already no parameter in the x or y value. </p><p>e.g. (p + q, 3) </p><p>locus is y = 3 </p><p>b) Type 2: obvious relationship between the values, usually only one </p><p>parameter. </p><p> 1,6 e.g. 2 tt</p><p>6</p><p>6</p><p>xt</p><p>tx</p><p>136</p><p>1</p><p>2</p><p>2</p><p>xy</p><p>ty</p><p>Eliminate the parameter (get rid of the ps and qs) </p></li><li><p>c) Type 3: not an obvious relationship between the values, use a </p><p>previously proven relationship between the parameters. </p></li><li><p>c) Type 3: not an obvious relationship between the values, use a </p><p>previously proven relationship between the parameters. </p><p> qpqp , e.g. 22</p><p>Given that pq = 3 </p></li><li><p>c) Type 3: not an obvious relationship between the values, use a </p><p>previously proven relationship between the parameters. </p><p> qpqp , e.g. 22</p><p>Given that pq = 3 pqqp</p><p>qpx</p><p>22</p><p>22</p></li><li><p>c) Type 3: not an obvious relationship between the values, use a </p><p>previously proven relationship between the parameters. </p><p> qpqp , e.g. 22</p><p>Given that pq = 3 pqqp</p><p>qpx</p><p>22</p><p>22</p><p>62 yx</p></li><li><p>c) Type 3: not an obvious relationship between the values, use a </p><p>previously proven relationship between the parameters. </p><p> qpqp , e.g. 22</p><p>Given that pq = 3 pqqp</p><p>qpx</p><p>22</p><p>22</p><p>62 yx</p><p>2005 Extension 1 HSC Q4c) </p><p>The points and lie on the parabola </p><p>The equation of the normal to the parabola at P is </p><p>and the equation of the normal at Q is given by </p><p> 2,2 apapP 2,2 aqaqQ ayx 42 32 apappyx </p><p>32 aqaqqyx </p></li><li><p>c) Type 3: not an obvious relationship between the values, use a </p><p>previously proven relationship between the parameters. </p><p> qpqp , e.g. 22</p><p>Given that pq = 3 pqqp</p><p>qpx</p><p>22</p><p>22</p><p>62 yx</p><p>2005 Extension 1 HSC Q4c) </p><p>The points and lie on the parabola </p><p>The equation of the normal to the parabola at P is </p><p>and the equation of the normal at Q is given by </p><p> 2,2 apapP 2,2 aqaqQ ayx 42 32 apappyx </p><p>32 aqaqqyx </p><p>(i) Show that the normals at P and Q intersect at the point R whose </p><p>coordinates are 2, 22 qpqpaqpapq</p></li><li><p>c) Type 3: not an obvious relationship between the values, use a </p><p>previously proven relationship between the parameters. </p><p> qpqp , e.g. 22</p><p>Given that pq = 3 pqqp</p><p>qpx</p><p>22</p><p>22</p><p>62 yx</p><p>2005 Extension 1 HSC Q4c) </p><p>The points and lie on the parabola </p><p>The equation of the normal to the parabola at P is </p><p>and the equation of the normal at Q is given by </p><p> 2,2 apapP 2,2 aqaqQ ayx 42 32 apappyx </p><p>32 aqaqqyx </p><p>(i) Show that the normals at P and Q intersect at the point R whose </p><p>coordinates are 2, 22 qpqpaqpapq</p><p>(ii) The equation of the chord PQ is . If PQ passes </p><p>through (0,a), show that pq = 1 </p><p> apqxqpy 2</p><p>1</p></li><li><p>(iii) Find the locus of R if PQ passes through (0,a) </p></li><li><p>(iii) Find the locus of R if PQ passes through (0,a) </p><p> qpapqx </p></li><li><p>(iii) Find the locus of R if PQ passes through (0,a) </p><p> qpapqx </p><p>a</p><p>xqp</p><p>qpax</p></li><li><p>(iii) Find the locus of R if PQ passes through (0,a) </p><p> qpapqx </p><p>a</p><p>xqp</p><p>qpax</p><p> 222 qpqpay</p></li><li><p>(iii) Find the locus of R if PQ passes through (0,a) </p><p> qpapqx </p><p>a</p><p>xqp</p><p>qpax</p><p> 222 qpqpay</p><p> 22 pqqpay</p></li><li><p>(iii) Find the locus of R if PQ passes through (0,a) </p><p> qpapqx </p><p>a</p><p>xqp</p><p>qpax</p><p> 222 qpqpay</p><p> 22 pqqpay</p><p> 21</p><p>2</p><p>2</p><p>a</p><p>xay</p></li><li><p>(iii) Find the locus of R if PQ passes through (0,a) </p><p> qpapqx </p><p>a</p><p>xqp</p><p>qpax</p><p> 222 qpqpay</p><p> 22 pqqpay</p><p> 21</p><p>2</p><p>2</p><p>a</p><p>xay</p><p>aa</p><p>xy 3</p><p>2</p></li><li><p>2004 Extension 1 HSC Q4b) </p><p>The two points and are on the parabola 2,2 apapP 2,2 aqaqQ ayx 42 </p></li><li><p>2004 Extension 1 HSC Q4b) </p><p>The two points and are on the parabola 2,2 apapP 2,2 aqaqQ ayx 42 </p><p>2attxy (i) The equation of the tangent to at an arbitrary point </p><p>on the parabola is </p><p> Show that the tangents at the points P and Q meet at R, where R is </p><p>the point </p><p>ayx 42 2,2 atat</p><p> apqqpa ,</p></li><li><p>2004 Extension 1 HSC Q4b) </p><p>The two points and are on the parabola 2,2 apapP 2,2 aqaqQ ayx 42 </p><p>2attxy (i) The equation of the tangent to at an arbitrary point </p><p>on the parabola is </p><p> Show that the tangents at the points P and Q meet at R, where R is </p><p>the point </p><p>(ii) As P varies, the point Q is chosen so that is a right angle, </p><p>where O is the origin. </p><p> Find the locus of R. </p><p>POQ</p><p>ayx 42 2,2 atat</p><p> apqqpa ,</p></li><li><p>2004 Extension 1 HSC Q4b) </p><p>The two points and are on the parabola 2,2 apapP 2,2 aqaqQ ayx 42 </p><p>2attxy (i) The equation of the tangent to at an arbitrary point </p><p>on the parabola is </p><p> Show that the tangents at the points P and Q meet at R, where R is </p><p>the point </p><p>(ii) As P varies, the point Q is chosen so that is a right angle, </p><p>where O is the origin. </p><p> Find the locus of R. </p><p>POQ</p><p>ayx 42 2,2 atat</p><p> apqqpa ,</p><p>1 OQOP mm</p></li><li><p>2004 Extension 1 HSC Q4b) </p><p>The two points and are on the parabola 2,2 apapP 2,2 aqaqQ ayx 42 </p><p>2attxy (i) The equation of the tangent to at an arbitrary point </p><p>on the parabola is </p><p> Show that the tangents at the points P and Q meet at R, where R is </p><p>the point </p><p>(ii) As P varies, the point Q is chosen so that is a right angle, </p><p>where O is the origin. </p><p> Find the locus of R. </p><p>POQ</p><p>ayx 42 2,2 atat</p><p> apqqpa ,</p><p>1 OQOP mm</p><p>102</p><p>0</p><p>02</p><p>0 22</p><p>aq</p><p>aq</p><p>ap</p><p>ap</p></li><li><p>2004 Extension 1 HSC Q4b) </p><p>The two points and are on the parabola 2,2 apapP 2,2 aqaqQ ayx 42 </p><p>2attxy (i) The equation of the tangent to at an arbitrary point </p><p>on the parabola is </p><p> Show that the tangents at the points P and Q meet at R, where R is </p><p>the point </p><p>(ii) As P varies, the point Q is chosen so that is a right angle, </p><p>where O is the origin. </p><p> Find the locus of R. </p><p>POQ</p><p>ayx 42 2,2 atat</p><p> apqqpa ,</p><p>1 OQOP mm</p><p>102</p><p>0</p><p>02</p><p>0 22</p><p>aq</p><p>aq</p><p>ap</p><p>ap</p><p>pqaqap 222 4</p></li><li><p>2004 Extension 1 HSC Q4b) </p><p>The two points and are on the parabola 2,2 apapP 2,2 aqaqQ ayx 42 </p><p>2attxy (i) The equation of the tangent to at an arbitrary point </p><p>on the parabola is </p><p> Show that the tangents at the points P and Q meet at R, where R is </p><p>the point </p><p>(ii) As P varies, the point Q is chosen so that is a right angle, </p><p>where O is the origin. </p><p> Find the locus of R. </p><p>POQ</p><p>ayx 42 2,2 atat</p><p> apqqpa ,</p><p>1 OQOP mm</p><p>102</p><p>0</p><p>02</p><p>0 22</p><p>aq</p><p>aq</p><p>ap</p><p>ap</p><p>pqaqap 222 4</p><p>4pq</p></li><li><p>2004 Extension 1 HSC Q4b) </p><p>The two points and are on the parabola 2,2 apapP 2,2 aqaqQ ayx 42 </p><p>2attxy (i) The equation of the tangent to at an arbitrary point </p><p>on the parabola is </p><p> Show that the tangents at the points P and Q meet at R, where R is </p><p>the point </p><p>(ii) As P varies, the point Q is chosen so that is a right angle, </p><p>where O is the origin. </p><p> Find the locus of R. </p><p>POQ</p><p>ayx 42 2,2 atat</p><p> apqqpa ,</p><p>1 OQOP mm</p><p>102</p><p>0</p><p>02</p><p>0 22</p><p>aq</p><p>aq</p><p>ap</p><p>ap</p><p>pqaqap 222 4</p><p>4pq</p><p>apqy </p></li><li><p>2004 Extension 1 HSC Q4b) </p><p>The two points and are on the parabola 2,2 apapP 2,2 aqaqQ ayx 42 </p><p>2attxy (i) The equation of the tangent to at an arbitrary point </p><p>on the parabola is </p><p> Show that the tangents at the points P and Q meet at R, where R is </p><p>the point </p><p>(ii) As P varies, the point Q is chosen so that is a right angle, </p><p>where O is the origin. </p><p> Find the locus of R. </p><p>POQ</p><p>ayx 42 2,2 atat</p><p> apqqpa ,</p><p>1 OQOP mm</p><p>102</p><p>0</p><p>02</p><p>0 22</p><p>aq</p><p>aq</p><p>ap</p><p>ap</p><p>pqaqap 222 4</p><p>4pq</p><p>apqy </p><p>ay 4</p></li><li><p>2004 Extension 1 HSC Q4b) </p><p>The two points and are on the parabola 2,2 apapP 2,2 aqaqQ ayx 42 </p><p>2attxy (i) The equation of the tangent to at an arbitrary point </p><p>on the parabola is </p><p> Show that the tangents at the points P and Q meet at R, where R is </p><p>the point </p><p>(ii) As P varies, the point Q is chosen so that is a right angle, </p><p>where O is the origin. </p><p> Find the locus of R. </p><p>POQ</p><p>ayx 42 2,2 atat</p><p> apqqpa ,</p><p>1 OQOP mm</p><p>102</p><p>0</p><p>02</p><p>0 22</p><p>aq</p><p>aq</p><p>ap</p><p>ap</p><p>pqaqap 222 4</p><p>4pq</p><p>apqy </p><p>ay 4Exercise 9J; odds </p><p>up to 23 </p></li></ul>