11X1 T12 09 locus problems

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<ul><li><p>Locus Problems</p></li><li><p>Locus Problems(get rid of the ps and qs)</p></li><li><p>Locus Problems(get rid of the ps and qs)</p><p>We find the locus of a point.</p></li><li><p>Locus Problems(get rid of the ps and qs)</p><p>We find the locus of a point.</p><p>1 Find the coordinates of the point whose locus you are finding.</p></li><li><p>Locus Problems(get rid of the ps and qs)</p><p>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 (i.e. get rid of the parameters)</p></li><li><p>Locus Problems(get rid of the ps and qs)</p><p>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 (i.e. get rid of the parameters)</p><p>a) Type 1: already no parameter in the x or y value.</p></li><li><p>Locus Problems(get rid of the ps and qs)</p><p>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 (i.e. get rid of the parameters)</p><p>a) Type 1: already no parameter in the x or y value.e.g. (p + q, 3) </p></li><li><p>Locus Problems(get rid of the ps and qs)</p><p>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 (i.e. get rid of the parameters)</p><p>a) Type 1: already no parameter in the x or y value.e.g. (p + q, 3) locus is y = 3 </p></li><li><p>Locus Problems(get rid of the ps and qs)</p><p>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 (i.e. get rid of the parameters)</p><p>a) Type 1: already no parameter in the x or y value.e.g. (p + q, 3) locus is y = 3 </p><p>b) Type 2: obvious relationship between the values, usually only one parameter.</p></li><li><p>Locus Problems(get rid of the ps and qs)</p><p>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 (i.e. get rid of the parameters)</p><p>a) Type 1: already no parameter in the x or y value.e.g. (p + q, 3) locus is y = 3 </p><p>b) Type 2: obvious relationship between the values, usually only one parameter. 1,6 e.g. 2 tt</p></li><li><p>Locus Problems(get rid of the ps and qs)</p><p>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 (i.e. get rid of the parameters)</p><p>a) Type 1: already no parameter in the x or y value.e.g. (p + q, 3) locus is y = 3 </p><p>b) Type 2: obvious relationship between the values, usually only one parameter. 1,6 e.g. 2 tt</p><p>6</p><p>6xt</p><p>tx</p></li><li><p>Locus Problems(get rid of the ps and qs)</p><p>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 (i.e. get rid of the parameters)</p><p>a) Type 1: already no parameter in the x or y value.e.g. (p + q, 3) locus is y = 3 </p><p>b) Type 2: obvious relationship between the values, usually only one parameter. 1,6 e.g. 2 tt</p><p>6</p><p>6xt</p><p>tx</p><p>136</p><p>12</p><p>2</p><p>xy</p><p>ty</p></li><li><p>c) Type 3: not an obvious relationship between the values, use a previously proven relationship between the parameters.</p></li><li><p>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</p></li><li><p>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</p><p>qpx</p><p>2222</p></li><li><p>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</p><p>qpx</p><p>2222</p><p>62 yx</p></li><li><p>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</p><p>qpx</p><p>2222</p><p>62 yx</p><p>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 </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 previously proven relationship between the parameters. qpqp , e.g. 22Given that pq = 3 pqqp</p><p>qpx</p><p>2222</p><p>62 yx</p><p>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 </p><p> 2,2 apapP 2,2 aqaqQ ayx 42 32 apappyx </p><p>32 aqaqqyx (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 previously proven relationship between the parameters. qpqp , e.g. 22Given that pq = 3 pqqp</p><p>qpx</p><p>2222</p><p>62 yx</p><p>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 </p><p> 2,2 apapP 2,2 aqaqQ ayx 42 32 apappyx </p><p>32 aqaqqyx (i) Show that the normals at P and Q intersect at the point R whose </p><p>coordinates are 2, 22 qpqpaqpapq(ii) The equation of the chord PQ is . If PQ passes </p><p>through (0,a), show that pq = 1 apqxqpy </p><p>21</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>axqp</p><p>qpax</p></li><li><p>(iii) Find the locus of R if PQ passes through (0,a) </p><p> qpapqx </p><p>axqp</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>axqp</p><p>qpax</p><p> 222 qpqpay 22 pqqpay</p></li><li><p>(iii) Find the locus of R if PQ passes through (0,a) </p><p> qpapqx </p><p>axqp</p><p>qpax</p><p> 222 qpqpay 22 pqqpay</p><p> 2122</p><p>axay</p></li><li><p>(iii) Find the locus of R if PQ passes through (0,a) </p><p> qpapqx </p><p>axqp</p><p>qpax</p><p> 222 qpqpay 22 pqqpay</p><p> 2122</p><p>axay</p><p>aaxy 3</p><p>2</p></li><li><p>2004 Extension 1 HSC Q4b)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)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 Show that the tangents at the points P and Q meet at R, where R is the point</p><p>ayx 42 2,2 atat apqqpa ,</p></li><li><p>2004 Extension 1 HSC Q4b)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 Show that the tangents at the points P and Q meet at R, where R is the point</p><p>(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. </p><p>POQ</p><p>ayx 42 2,2 atat apqqpa ,</p></li><li><p>2004 Extension 1 HSC Q4b)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 Show that the tangents at the points P and Q meet at R, where R is the point</p><p>(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. </p><p>POQ</p><p>ayx 42 2,2 atat apqqpa ,</p><p>1 OQOP mm</p></li><li><p>2004 Extension 1 HSC Q4b)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 Show that the tangents at the points P and Q meet at R, where R is the point</p><p>(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. </p><p>POQ</p><p>ayx 42 2,2 atat apqqpa ,</p><p>1 OQOP mm1</p><p>020</p><p>020 22 </p><p>aqaq</p><p>apap</p></li><li><p>2004 Extension 1 HSC Q4b)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 Show that the tangents at the points P and Q meet at R, where R is the point</p><p>(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. </p><p>POQ</p><p>ayx 42 2,2 atat apqqpa ,</p><p>1 OQOP mm1</p><p>020</p><p>020 22 </p><p>aqaq</p><p>apap</p><p>pqaqap 222 4</p></li><li><p>2004 Extension 1 HSC Q4b)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 Show that the tangents at the points P and Q meet at R, where R is the point</p><p>(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. </p><p>POQ</p><p>ayx 42 2,2 atat apqqpa ,</p><p>1 OQOP mm1</p><p>020</p><p>020 22 </p><p>aqaq</p><p>apap</p><p>pqaqap 222 44pq</p></li><li><p>2004 Extension 1 HSC Q4b)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 Show that the tangents at the points P and Q meet at R, where R is the point</p><p>(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. </p><p>POQ</p><p>ayx 42 2,2 atat apqqpa ,</p><p>1 OQOP mm1</p><p>020</p><p>020 22 </p><p>aqaq</p><p>apap</p><p>pqaqap 222 44pq</p><p>apqy </p></li><li><p>2004 Extension 1 HSC Q4b)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 Show that the tangents at the points P and Q meet at R, where R is the point</p><p>(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. </p><p>POQ</p><p>ayx 42 2,2 atat apqqpa ,</p><p>1 OQOP mm1</p><p>020</p><p>020 22 </p><p>aqaq</p><p>apap</p><p>pqaqap 222 44pq</p><p>apqy ay 4</p></li><li><p>2004 Extension 1 HSC Q4b)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 Show that the tangents at the points P and Q meet at R, where R is the point</p><p>(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. </p><p>POQ</p><p>ayx 42 2,2 atat apqqpa ,</p><p>1 OQOP mm1</p><p>020</p><p>020 22 </p><p>aqaq</p><p>apap</p><p>pqaqap 222 44pq</p><p>apqy ay 4</p><p>Exercise 9J; oddsup to 23</p><p>Slide Number 1Slide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33Slide Number 34Slide Number 35Slide Number 36</p></li></ul>