Stright line grphs, Mied Eercise Grdient m ( y ),,, The eqution of the line is: y m( ) ( ) + y + Sustitute (k, ) into y + k + k k Multiply ech side y : k k The grdient of AB is: y y So: ( k ) 8 k k 8 k k 8 k Multiply ech side y (8 k): k ( 8 k) Multiply ech term y : k 8 k k 8 k k k. So A nd B hve coordintes (, ) nd (8, ). The eqution of the line is: y y y 8 Multiply ech side y : ( ) y + The eqution of L is: y m ( ) y + The eqution of L is: y ( ) 8 ( ) 8 + y + Solve y + nd y + simultneously. + + 8 + 8 8 9 Sustitute 9 into y + : y 9 + The lines intersect t C(9, ). Person Eduction Ltd. Copying permitted for purchsing institution only. This mteril is not copyright free.
The eqution of l is: y y y y Multiply ech side y : ( ) y y ( ) y Solve + y nd y simultneously. Sustitute: + ( ) 9 9 + 9 9 9 Sustitute into y : y 9 The coordintes of C re (, ). (, y) (, ), (, y) ( 9, 9) The eqution of L is: y y y 9 9 Multiply ech side y : ( ) Multiply ech term y : ( ) y + 9 y+ 9 The eqution of L is y + 9. (, y) (, ), (, y) (, ) The eqution of l is: y y y Multiply ech side y : Note: + + The eqution of l is: y m ( 9) 9 y + y + 9 A(, ), B ( +, + ) The grdient of the line through A nd B is: y y + + + + Person Eduction Ltd. Copying permitted for purchsing institution only. This mteril is not copyright free.
Rtionlising the denomintor: (+ ) ( ) + (+ ) ( ) The eqution of the line is: y + c Sustituting nd y into y + c: + c c The eqution of line l is: y + Line l meets the -is when y. When y,. C is the point (, ). 8 A(, ), B(, 8) The grdient of AB is: y y 8 The grdient of line perpendiculr to AB is: The eqution of p is: y m ( ) 8 ( ) 8 + y + Sustitute in the eqution for AB: y () + The coordintes of C re (, ). 9 The line psses through A(, ) nd is perpendiculr to l: y. y y The grdient of y is. The grdient of line perpendiculr to y is. The eqution of the line m is: y m ( ) ( ) y + Or, since A is y-intercept, the eqution cn e written once the +. grdient is known i.e y To find P, solve y + nd y simultneously. Sustitute: ( + ) + Sustitute into y + : y + + The lines intersect t P(, ), s required. c A line prllel to the line m hs grdient. The eqution of the line n is: y m ( ) ( ) y + Person Eduction Ltd. Copying permitted for purchsing institution only. This mteril is not copyright free.
9 c To find Q solve y nd y + simultneously. Sustitute: + + Sustitute into y + : y + + The lines intersect t Q(, ). A(, ) nd B(, ) The grdient of the line through A nd B is: y y ( ) 9 The eqution of the line through A nd B is: y + c Sustituting nd y into y + c: () + c so c As in Q9, the point A is the y-intercept so the eqution cn e written once the grdient hs een clculted. l: y l: + y 8 To find point D, solve simultneously y sustituting l into l. + 8 when, + y 8, y D is the point (, ). The se of the tringle AC is units. The height of the tringle is units. Are ACD is units A(, ) nd B(, ) The eqution of l through A nd B is: y y y Multiply ech side y : ( ) Note: + y + + y The eqution of l through C(, ) with grdient is: y m ( ) ( ) ( ) + + y + Person Eduction Ltd. Copying permitted for purchsing institution only. This mteril is not copyright free.
A(, ), B(, ) nd C(k, ) The grdient of AB is: y y ( ) ( ) 8 Since ABC is right ngle the grdient of BC is: y y So k k Multiply ech side y (k ): ( k ) k + k k c The eqution of the line pssing through B nd C is: y y y Multiply ech side y : ( ) + y + + y + d Rememer ngle B is right ngle. Use the digrm or the distnce formul to find the lengths AB nd BC. AB 8 + 8 BC + Are of ABC 8 units The eqution of the line through (, ) nd (, ) is: y y y y ( ) ( ) + Multiply ech side y : ( ) ( + ) ( Note: nd ) + + 8 Person Eduction Ltd. Copying permitted for purchsing institution only. This mteril is not copyright free.
For the coordintes of A, sustitute y : + 8 8 8 8 8 The coordintes of A re (, ). For the coordintes of B, sustitute : + y 8 y 8 y 8 y 8 The coordintes of B re 8 The re of OAB is: 8 8,. Solve y + nd y simultneously. Sustitute: ( ) + 8 + 9 9 9 Now sustitute into y : y ( ) 8,. The coordintes of A re Rerrnge l: y + into the form y m + c: y y l hs grdient nd it meets the coordinte es t (, ). l hs grdient nd it meets the y-is t (, ). l meets the -is when y. Sustitute y into the eqution: l meets the -is t (, ). c The grdient of l is. The grdient of line perpendiculr to l is. The eqution of this line is: y m ( ) y+ y Multiply ech term y : y The eqution of the line is y. Person Eduction Ltd. Copying permitted for purchsing institution only. This mteril is not copyright free.
A(, ) nd B(, ) The grdient of the line l through A nd B is: y y 8 The eqution of l is: y + c Sustituting nd y into y + c: () + c c 8 y + c + y The grdient of the line l is, the y- intercept is. y c Solve + y nd y simultneously. + ( ) 8 When 8: y ( 8) y C is the point ( 8, ). d The grdient of OA is: y y d The grdient of OC is: y y ( 8). e Therefore the lines OA nd OC re perpendiculr. OA ( ) + ( ) (( 8) ) + ( ) 8 f Are of OAB 8 units (, ) nd (, ) The distnce d etween the points is: d ( ) + ( y y ) (( ) ) + ( ) 9 + For points (, ) nd (, ),. Sustitute into. Distnce c For points (, ) nd ( 9, ),. Sustitute into. Distnce Person Eduction Ltd. Copying permitted for purchsing institution only. This mteril is not copyright free.
d For points (, ) nd (, ),. Sustitute into. Distnce (, y) is point on y, so its coordintes re (, ). The distnce etween A(, ) nd (, ) is: d ( ) + ( y y ) ( ( )) + ( ) d 9 + When : y ( ) The point is (, ) e + + + + + 9 8 8+ 8+ 8 8 ( + )( ) or 8 When, y ( ) When, y () 8 The points re B(, ) nd C(, ). c The grdient of the line y is, so the perpendiculr line hs grdient. Its eqution is: y + c When nd y : ( ) + c c y + d Solving y + simultneously: + nd y BC ( ) + ( y y ) ( ( )) + ( ( )) + 8 8 Distnce from A(, ) to (, ) is: ( ( )) + ( ) + ( ) Are of tringle is:.8 units 8 Grdient of the line of est fit is: y y 9. Person Eduction Ltd. Copying permitted for purchsing institution only. This mteril is not copyright free. 8
8 C P +. Using the point (, ):.() + C.P c is the grdient, which is the increse in cron dioide emissions in millions of tonnes for every million tonnes of oil pollution. d The model is not vlid for smll vlues of P s negtive mount of cron dioide emissions is not possile. Chllenge (, ), B(, 8) nd C(, ) The eqution of AB is: y y y ( ) ( ) 8 ( ) ( ) y+ + y + + y y The grdient of AB. The grdient of line perpendiculr to AB. The eqution of the perpendiculr to AB through C(, ) is: y ( ( )) y y + 8 Point D is where the line nd the perpendiculr intersect. Solve the equtions y nd y + 8 simultneously. + 8 Multiply ech term y. 9 + 8 Now sustitute into y + 8: y () + 8 y D is the point (, ). Person Eduction Ltd. Copying permitted for purchsing institution only. This mteril is not copyright free. 9
Chllenge AB ( ) + ( y y ) ( ( )) + (8 ( )) + CD ( ( )) + ( ) 8 + ( ) 8 Are of ABC 8 units A(, 8), B(9, 9) nd C(, ) The grdient of AB is: y y 9 8 9 l is perpendiculr to AB, so its grdient is. It psses through C, so its eqution is: y + c () + c c The eqution of l is y +. The grdient of BC is: y y 9 9 l is perpendiculr to BC, so its grdient is. It psses through A, so its eqution is: y + c 8 () + c c 8 The eqution of l is y + 8. The grdient of AC is: y y 8 l is perpendiculr to BC, so its grdient is. It psses through B, so its eqution is: y + c 9 (9) + c c The eqution of l is y +. Solve l nd l simultneously. + + 8 + + 8 8 8 9 y ( 8 9 ) + 9 Their point of intersection is ( 8 9, 9 ). Now solve l nd l simultneously. + 8 + + + 9 8 8 9 y ( 8 ) + 9 9 Their point of intersection is ( 8, ). 9 9 Therefore, l, l nd l ll intersect t ( 8, ). 9 9 A(, ), B(, ) nd C(c, ) The grdient of AB is: y y l is perpendiculr to AB so its grdient is. Person Eduction Ltd. Copying permitted for purchsing institution only. This mteril is not copyright free.
It psses through C so its eqution is: y + k where k is the y-intercept. At C, c nd y. c + k c k Now solve l nd l simultneously. c ( ) y The intersection of l nd l is the point, c. Therefore, l, l nd l ll intersect t, c. The eqution of line l is: c y + The grdient of BC is: y y c c l is perpendiculr to BC so its grdient is c. It psses through A, so its eqution is: c y + K where K is the y-intercept. At A,, y. c () + K K The eqution of line l is c y. l is the verticl line through (, ), so its eqution is. Solve l nd l simultneously. c y + c ( ) The intersection of l nd l is the point c ( ) (, ). Person Eduction Ltd. Copying permitted for purchsing institution only. This mteril is not copyright free.