1) Find a vector parameterization for the line that passes through P(3 ...
1 coordinate geometry - Maths Excel Class · 3. Line 1 passes through the points (6 ,9) and (5 ,3)....
Transcript of 1 coordinate geometry - Maths Excel Class · 3. Line 1 passes through the points (6 ,9) and (5 ,3)....
Co-ordinateGeometry–Co-ordinates
©MathsExcelClass2015 3rdYrMary1
Everypointhastwoco-ordinates.
(3, 2)
Plotthefollowingpointsontheplane.
A(4, 1) D(2, −5) G(6, 3)B(3, 3) E(−4, −4) H(6, −3)C(−6, 5) F(−1, 2) I(−3, 1.5)
𝑥co-ordinate 𝑦co-ordinate
. (3, 2)
Co-ordinate Geometry – Gradient
©MathsExcelClass2015 3rdYrMary2
Thegradient(𝑚)istheslopeoftheline.Positivegradient(𝒎 > 𝟎) Negativegradient(𝒎 < 𝟎)
𝒎 = 𝟎
Abiggergradientmakesasteeperline.
𝑚 = 1
𝑚 = 2
𝑚 = −1𝑚 = −2
Co-ordinate Geometry – Gradient
©MathsExcelClass2015 3rdYrMary3
Example1Findthegradientofthefollowingline.
𝑚 = 56
Example2Findthegradientoftheline.
𝑚 = 7897:;89;:
= <9=
>95
= ?
=
𝑚 = @ABC@DE
= 7897:
;89;:
.
(6, 7).
(3, 2)
Rise = 7 − 2 = 5
Run = 6 − 3 = 3
. (4, 3)
Rise=3
Run=4
Co-ordinate Geometry – Gradient
©MathsExcelClass2015 3rdYrMary4
QuestionsPlotthefollowingpointsandfindthegradient.
1. (5, 3)and(2, 1)
2. (7, 5)and(1, 1)
3. (3, 8)and(0, 0)
4. (5, 4)and(−2, 1)
5. (−3, −4)and(9, 1)
6. (−4, −1)and(−8, −2)
Findthegradientofthelineforeachpairofpoints.
1. (5, 8)and(1, 4)
2. (−3, 5)and(10, −2)
3. (10, 35)and(2, 3)
4. (−4, 7)and(−5, 1)
5. (22, 7)and(2, 2)
Co-ordinateGeometry–Equationofaline
©MathsExcelClass2015 3rdYrMary5
Therearetwoformsfortheequationofaline.Generalform: 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0Gradient–Interceptform: 𝑦 = 𝑚𝑥 + 𝑏Therearethreewaystofindtheequationofalinedependingonwhatinformationyouhave.
1. Gradientandpoint
2. Twopoints
3. Gradientand𝑦-intercept TestingifapointisonthelineSubthepointintotheequation.Ifitsatisfiestheequation,thepointliesontheline.ExampleDoesthepoint(3,4)lieontheline2𝑥 − 3𝑦 + 6 = 0?LHS = 2 3 − 3 4 + 6 = 6 − 12 + 6 = 0 =RHS∴Thepoint(3,4)isontheline.
gradient 𝑦-intercept
𝒚 − 𝒚𝟏 = 𝒎(𝒙 − 𝒙𝟏)
𝒚 − 𝒚𝟏𝒙 − 𝒙𝟏
=𝒚𝟐 − 𝒚𝟏𝒙𝟐 − 𝒙𝟏
𝒚 = 𝒎𝒙 + 𝒃
𝑥
𝑦
Co-ordinateGeometry–Equationofaline
©MathsExcelClass2015 3rdYrMary6
𝒙-interceptsand𝒚-intercepts The𝒚-interceptis3.Theco-ordinatesare(0, 3).The𝒙-interceptis−4.Theco-ordinatesare(−4, 0).Example1Theequationofalineis3𝑥 − 4𝑦 + 12 = 0.Findthe𝑥-interceptand𝑦-intercept.𝒚-interceptSub𝑥 = 0intotheequation.3𝑥 − 4𝑦 + 12 = 03(𝟎) − 4𝑦 + 12 = 04𝑦 = 12𝑦 = 3 ∴The𝑦-interceptis3.𝒙-interceptSub𝑦 = 0intotheequation.3𝑥 − 4𝑦 + 12 = 03𝒙 − 4(𝟎) + 12 = 03𝑥 = −12𝑥 = −4 ∴The𝑥-interceptis−4.
𝑥
𝑦
3 −
−4
−
Co-ordinateGeometry–Equationofaline
©MathsExcelClass2015 3rdYrMary7
Findthe𝑥-interceptand𝑦-interceptofthefollowinglines.Sketchthegraphs.
1. 5𝑥 − 2𝑦 + 10 = 0
2. 3𝑥 − 2𝑦 + 6 = 0
3. 10𝑥 − 𝑦 + 10 = 0
4. 2𝑦 − 7𝑥 + 14 = 0
5. 𝑦 − 2𝑥 + 6 = 0
6. 𝑦 = T=𝑥 + 4
7. 𝑦 = =5𝑥 − 12
8. 𝑦 = 3𝑥 − 11
Co-ordinateGeometry–Equationofaline
©MathsExcelClass2015 3rdYrMary8
Findtheequationofthefollowinglinesingradient-interceptform.
1. Thelinehasagradientof2andpassesthroughthepoint(3, 4).
2. Thelinehasagradientof7andpassesthroughthepoint(−5, 2).
3. ThelinehasagradientofT=andpassesthroughthepoint(1, 4).
4. Thelinepassesthroughthepoints(1, 3)and(6, 8).
5. Thelinepassesthroughthepoints(6, 2)and(9, 3).
6. Thelinehasagradientof5?anda𝑦-interceptof4.
Co-ordinateGeometry–Equationofaline
©MathsExcelClass2015 3rdYrMary9
Findtheequationofthefollowinglinesingeneralform.
1. Thelinehasagradientof?>andpassesthroughthepoint(1, 2).
2. Thelinehasagradientof=Uandpassesthroughthepoint(5, −3).
3. Thelinepassesthroughthepoints(3, −1)and(5, 6).
4. Thelinepassesthroughthepoints(2, 11)and(−3, −1).
5. ThelinehasagradientofT6andan𝑥-interceptof8.
Co-ordinateGeometry–Equationofaline
©MathsExcelClass2015 3rdYrMary10
1. Showthatthepoint(0, 2)liesontheline𝑥 − 4𝑦 + 8 = 0.
2. Showthatthepoint(−2, 4)liesontheline𝑦 = 8𝑥 + 20.
3. Showthatthepoint(1, −7)liesontheline𝑦 = 4𝑥 − 11.
Co-ordinate Geometry – Sketching
©MathsExcelClass2015 3rdYrMary11
Fortheequation𝒚 = 𝟐𝒙 + 𝟑fillinthetableofvalues.𝑦
𝑥 −4 −3 −2 −1 0 1 2 3 4
Sketchthegraph.Remember:
• Alwayslabelthe𝑥and𝑦axeswithbothletterandarrow
• Alwaysputanarrowateachendofthelinesketched
• Labelthe𝑥-interceptand𝑦-interceptifpossibleTosketchagraph,youonlyneedtoknowtwopoints.Atableofvaluesisnotnecessary.Changingtheequationintogradient-interceptformisuseful.
Co-ordinate Geometry – Sketching
©MathsExcelClass2015 3rdYrMary12
Example1Sketchthelinethatpassesthroughthepoints(1, 4)and(5, 6).Plotthepoints. Example2Sketchthegraphof2𝑥 − 3𝑦 + 6 = 0.𝑥-intercept:−3𝑦-intercept:2Example3Sketchthegraphof𝑦 = 3
4 𝑥 + 2.𝑦-intercept:2Gradient=@ABC
@DE= 5
6
. . (1, 4)
(5, 6) . . (1, 4)
(5, 6)
−
−
2
−3
−
2𝑟𝑢𝑛 = 4
𝑟𝑖𝑠𝑒 = 3
. (4, 5)
−
2
. (4, 5)
Co-ordinate Geometry – Sketching
©MathsExcelClass2015 3rdYrMary13
1. Sketchthelinethatpassesthroughthepoints(3, 5)and(9, 7).Whatisthegradient?
2. Sketchthelinethatpassesthroughthepoints(−6, −5)and(2, 5).Whatisthegradient?
3. Sketchthelinethatpassesthroughthepoints(1, −8)and(10, 1).Whatisthegradient?
Co-ordinate Geometry – Sketching
©MathsExcelClass2015 3rdYrMary14
4. Graphthelinewithgradient=5anda𝑦-interceptof4.
5. Graphthelinewithgradient−52anda𝑦-interceptof1.
6. Graphthelinewithgradient]Uanda𝑦-interceptof−3.
Co-ordinate Geometry – Sketching
©MathsExcelClass2015 3rdYrMary15
7. Sketchthelinewithequation𝑦 = 14 𝑥 + 5.
8. Graphthelinearfunction𝑦 = −2𝑥 + 6.
9. Graphthelinearfunction𝑦 = −3𝑥 − 4.
Co-ordinate Geometry – Sketching
©MathsExcelClass2015 3rdYrMary16
10. Sketchthelinewithequation3𝑥 − 2𝑦 − 12 = 0.
11. Graphthelinearfunction8𝑥 + 3𝑦 − 24 = 0.
12. Graphthelinearfunction4𝑥 + 3𝑦 + 12 = 0.
Co-ordinate Geometry – Parallel lines
©MathsExcelClass2015 3rdYrMary17
Example1IsLine1paralleltoLine2?Line1passesthrough Line2passesthrough(2, 6)and(0, 1). (4, 5)and(2, 0).
𝑚T =6−12−0 =
?= 𝑚= =
5−04−2 =
?=
𝑚T = 𝑚= ∴Line1isparalleltoLine2.Example2Arethelinesparallel?Line1:𝑦 = 1
4 𝑥 + 6. Line2:𝑥 − 4𝑦 + 23 = 0 𝒎𝟏 =
𝟏𝟒 4𝑦 = 𝑥 + 23
𝑦 = 1
4 𝑥 +234
𝒎𝟐 =
𝟏𝟒
𝒎𝟏 = 𝒎𝟐∴Thelinesareparallel.
Parallellines:Gradientsarethesame
𝒎𝟏 = 𝒎𝟐
−1−
2
(4, 5). . (2, 6)Line1
Line2
Co-ordinate Geometry – Parallel lines
©MathsExcelClass2015 3rdYrMary18
1. Line1passesthroughthepoints(4, 5)and(2, 2).Line2passesthroughthepoints(8, 7)and(6, 4).Sketchbothlinesonthesameplaneandprovetheyareparallel.
2. LineAhastheequation𝑥 − 7𝑦 − 33 = 0.LineBisparalleltoLineAandpassesthroughthepoint(5, 1).FindtheequationofLineBandsketchbothlinesonthesameplane.
3. Provethefollowinglinesareparallelandsketcheachpaironthesameplane.
a. Line1:𝑦 = 45 𝑥 + 1
Line2:4𝑥 − 5𝑦 + 20 = 0
b. Line1:7𝑥 + 3𝑦 + 21 = 0
Line2:7𝑥 + 3𝑦 − 42 = 0
Co-ordinate Geometry – Perpendicular lines
©MathsExcelClass2015 3rdYrMary19
Perpendicularlinesintersectatanangleof90°.𝑨𝑩 ⊥ 𝑪𝑫.Example1IsLine1perpendiculartoLine2?Line1hasequation Line2hasequation4𝑥 − 3𝑦 + 3 = 0. 3𝑥 + 4𝑦 − 16 = 0.ChangetoG-Iform. 𝑦 = 4
3 𝑥 + 1 𝑦 = −34 𝑥 + 4
𝑚T= 43 𝑚== −3
4𝒎𝟏×𝒎𝟐 =
𝟒𝟑×− 𝟑
𝟒= −𝟏 ∴Line1isperpendiculartoLine2.
−1
−4
Line1:4𝑥 − 3𝑦 + 3 = 0
Line2:3𝑥 + 4𝑦 − 16 = 0
Perpendicularlines:Gradientsarethenegativereciprocalofeachother
𝒎𝟏 =−𝟏𝒎𝟐or𝒎𝟏𝒎𝟐 = −𝟏
A
BC
D
Co-ordinate Geometry – Perpendicular lines
©MathsExcelClass2015 3rdYrMary20
1. Writethenegativereciprocalofthefollowing:
a. =5
b. −>
<
c. 3
d. −5
2. Line1passesthroughthepoints(3, 7)and(1, 4).Line2passesthroughthepoints(3, 3)and(0, 5).ProvethatLine1isperpendiculartoLine2andsketchbothlinesonthesameplane.
3. Line1passesthroughthepoints(6, 9)and(5, 3).Line2passesthroughthepoints(8, 4)and(2, 5).ProvethatLine1isperpendiculartoLine2andsketchbothlinesonthesameplane.
4. LineAhastheequation2𝑥 − 5𝑦 + 1 = 0.LineBisperpendiculartoLineAandpassesthroughthepoint(3, 4).FindtheequationofLineBandsketchbothlinesonthesameplane.
Co-ordinate Geometry – Midpoint
©MathsExcelClass2015 3rdYrMary21
Midpoint:findtheaverageofeachco-ordinate.
𝑥co-ordinate=;:f;8= 𝑦co-ordinate=
7:f78=
Example1Findthemidpointbetween(1, 2)and(5, 8).Midpoint:
𝑥co-ordinate=Tf?= 𝑦co-ordinate=
=f]=
= 3 = 5∴Themidpointis(3, 5).
. (5, 8)
. (1, 2)
. (5, 8)
. (1, 2)
. (3, 5)
g𝑥T + 𝑥=2
,𝑦T + 𝑦=2
h
Co-ordinate Geometry – Midpoint
©MathsExcelClass2015 3rdYrMary22
Plotthefollowingpointsandfindthemidpoint.
1. (3, 4)and(7, 8)
2. 2, 6 and 4, 10
3. (5, 1)and(1, −9)
4. PointAisthe𝑦-interceptoftheline𝑦 = 2𝑥 + 3.PointBhasco-ordinates4, 11 .
a. ProvethatPointBliesontheline𝑦 = 2𝑥 + 3.
b. Sketchtheline𝑦 = 2𝑥 + 3indicatingPointAandPointB.FindthemidpointofAandBandaddittothediagram.
Co-ordinate Geometry – Distance
©MathsExcelClass2015 3rdYrMary23
ExampleFindthedistance(D)betweenthepoints(2, 4)and(6, 7).Disthehypotenuseofaright-angledtriangle.Theheightofthetriangleis(7 − 4).Thebaseofthetriangleis(6 − 2).UsingPythagoras’theorem:𝐷= = 6 − 2 = + 7 − 4 = 𝐷 = 6 − 2 = + 7 − 4 = 𝐷 = 4= + 3= = 16 + 9 = 25 𝐷 = 5𝑢𝑛𝑖𝑡𝑠
Distance= k(𝒙𝟐 − 𝒙𝟏)𝟐 + (𝒚𝟐 − 𝒚𝟏)𝟐units
. (6, 7).
(2, 4)D
. (6, 7).
(2, 4)Height= 7 − 4
Base= 6 − 2
D
Co-ordinate Geometry – Distance
©MathsExcelClass2015 3rdYrMary24
1. PointAhasco-ordinates 2, 7 andPointBhasco-ordinates 8, 15 .PlotthepointsandfindthedistanceoflineAB.
2. PointAhasco-ordinates 4, 1 andPointBhasco-ordinates 8, 7 .PlotthepointsandfindthedistanceoflineAB.
3. PointAhasco-ordinates 11, 10 andPointBhasco-ordinates 6, −2 .PlotthepointsandfindthedistanceoflineAB.
4. PointAhasco-ordinates −9, −7 andPointBhasco-ordinates −2, −6 .PlotthepointsandfindthedistanceoflineAB.
Co-ordinate Geometry – Distance
©MathsExcelClass2015 3rdYrMary25
5. PointAhasco-ordinates 4, 0 andPointBhasco-ordinates 6, −10 .PlotthepointsandfindthedistanceoflineAB.
6. PointAhasco-ordinates 2, 2 andPointBhasco-ordinates −3, 1 .PlotthepointsandfindthedistanceoflineAB.
7. Onamap,aredflagisplacedatco-ordinates −5, 8 andagreenflagisplacedatco-ordinates −8, −1 .Plotthepositionofthetwoflagsandfindthedifferencebetweenthem.
8. PointAisthemidpointbetween −3, 8 and 5, 4 .PointBhasco-ordinates 13, 12 .WhatisthedistanceoflineAB?
Co-ordinate Geometry – Distance
©MathsExcelClass2015 3rdYrMary26
9. PlotpointA −3, 2 ,pointB 5, 2 andpointC 5, 8 .
a. FindthedistanceofAB.
b. FindthedistanceofBC.
c. UsingeitherthedistanceformulaorPythagoras’theorem,findthedistanceofAC.
d. Findtheareaof△ 𝐴𝐵𝐶.
e. FindtheequationoflineAC.
f. FindtheequationsoflinesABandBC.
Co-ordinate Geometry – Perpendicular Distance
©MathsExcelClass2015 3rdYrMary27
Perpendiculardistanceistheshortestdistancebetweenapointandaline.Example1Findtheperpendiculardistancebetweenthepoint 4, 1 andtheline2𝑥 − 3𝑦 + 6 = 0.
Distance =𝑎𝑥1+𝑏𝑦1+𝑐
𝑎2+𝑏2
Fromtheequation2𝑥 − 3𝑦 + 6 = 0. 𝑎 = 2 𝑏 = −3 𝑐 = 6Fromthepoint 4, 1 𝑥T = 4 𝑦T = 1
Distance =2 4 −3 1 +6
22+(−3)2 =
8–3+64+9
=1113 = 11
13units
= 11 1313 units(rationaldenominator)
(4, 1).
2𝑥 − 3𝑦 + 6 = 0
D
Perpendiculardistance= |𝒂𝒙𝟏f𝒃𝒚𝟏f𝒄|k𝒂𝟐f𝒃𝟐
units
Co-ordinate Geometry – Perpendicular Distance
©MathsExcelClass2015 3rdYrMary28
Drawintheperpendiculardistancebetweenthefollowinglinesandpoints.
. .
.
. REMEMBER:
• Tousetheperpendiculardistanceformula,theequationofthelinemustbeingeneralform.
• Arrangetheequationsothatthe𝑥comesbeforethe𝑦.
Co-ordinate Geometry – Perpendicular Distance
©MathsExcelClass2015 3rdYrMary29
1. Findtheperpendiculardistancebetweenthepoint 1, 8 andtheline4𝑥 + 3𝑦 + 1 = 0.
2. Findtheperpendiculardistancebetweenthepoint 3, −2 andtheline8𝑥 + 6𝑦 − 5 = 0.
3. Findtheperpendiculardistancebetweenthepoint −5, 1 andtheline𝑥 − 7𝑦 − 10 = 0.
4. Findtheperpendiculardistancebetweenthepoint 3, 5 andtheline2𝑦 + 7𝑥 + 4 = 0.
Co-ordinate Geometry – Perpendicular Distance
©MathsExcelClass2015 3rdYrMary30
5. Findtheperpendiculardistancebetweenthepoint 4, 2 andtheline𝑦 =35 𝑥 + 3.
6. Sketchtheline𝑦 = −14 𝑥 − 2andfindtheperpendiculardistancefromthe
point 4, 2 totheline.
7. Sketchtheline𝑦 = 43 𝑥 − 11andfindtheperpendiculardistancefromthe
point 1, 2 totheline.