Analytical Geometry Notes (PDF)

Some important formulae from Gr 10 11 straight lines mkt C when youhave c y intercept Y find the equation y y MH Xi 61 y usefulto...

Some important formulae from Gr 10 11 straight lines mkt C when youhave c y intercept Y find the equation y y MH Xi 61 y usefulto when you have a point Blaig f Ala Point of A intersection use simultaneous equations horizontal line y c ffhf.fiythenAfub in vertical line Lanyvalue on x axis Distance Distance between Ax y and Baiya AB 6122.1 1929,7 AB 6122.1 1929,7 Gradient The gradient between Alx y and Baiya MAB small Angle of inclination no FIFTY thing O e if m 0 180 refc ftpffyte If tariff off 4 things to remember about gradient If m me lines are 11 M XM I lines are h MAB MBC ABC are collinear MAB tano m of horizontal line O m of vertical line undefined Midpoint The midpoint between Ax y and Baiya 11B I 91192  Ty It is the midpt of AC statement f go bcisadiamet.in i We can assume that ConV in semi circle reason prop of akite i if diameter 15 units AB BC then radius I 7,5 units MIERI M AE  rr why to I usepointAf5 3 given  The equation of a circle Centre radius equation form fffaffg.in k.at 19 bt f XStandard form r for a t ly b circles centred at circles centred off the origin the origin in am 61012 19 0 r xa ly b i r Vit g r be a tlyb F x̅ y  Example 1 Determine the equation of the circle1st with at the centre the origin and radius 7 22 y 82 i x 49 y b the centre the origin and passing through point 18 3 7C y p 187 1372 73 the centre 31 5 and radius B remembert x 9 19 b P 2 i x 372 9 1 57 B x 372 19 57 3 Homework Exercise 2.4 2.8 Equation of a circle with 2 4 Diameter CD where C 0 3 D 219 Midpt of CD centre of circle i Men 9 11 3 012 32 Hb i r 1 0 3 3 distance 37 keep as r because it is used in the formula 2 1 1 t ly 3 37 2.5 with x intepts 2,0 1 401 radius 5 we have r at least one point sub I Ij b 2 a 5 49 21 if i'Iii  From 49 21 89 9 i A 1 Sub at back into one of the original formulae 81 1 1112 b 9 4 i 1 1 4y 4 2 25 1 1 19 412 25 2.6 passing through the points 23 and 12 1 r.rs sub in 1231 2 r S 12 a 3 b 5 4 49 92 9 69 b 5 variables one side Elimination ftp9 p getrid of 49 Now sub in 12 1 r 55 2 2 a b2 2 a 1 b 5 5 i 8 an be 49C 2b from 8 6b 2b 8 8 sub into 411 8 i 99 56 i 9 329 17 O i 9 3 or a I 2 options 1 37 t ly 172 5 for eq OR x 11 1g 177 5 7.7 Passing through points 12 2 10 1 and 1110 sub 12 2 2 2 9 12 b p Eat 92 b f 49 4h 8 Sub foil 2 10 a2 11 b r 92 2b b r 1 92Ib b f I 92 b2 r2 2b 1 Ya co bi P p 92 b2 r2 29 1 From 2b 1 29 1 i a b Now from for 49 4h 8 29 1 but a b 49 49 8 29 1  Sub into 92 b r 29 1 but a I r 652 of circle Y Equation I 2.8 Passing through point 6 1 with circumference in it and with the centre on the axis y Sub in 6 1 and r 6 Circumference 6 0 11 b 2 62 HIIIII Egg 36 1 b Ñ we out the bracket it looks like this Ea uol.tcy m.ae 2 x y 1 36  CTS Find the equation centre radius by COMPLETING THE If we have x 112 19572 wecingultiply it out 49,9 x2 2 1y toy 25 9 i 71 274 9 Hoy 17 However if we are given the multiplied out form 71 274 9 Hoy 17 2 4 we can rewrite this in the form x a ly b 82 in order to find the centre radius 4 we do this by CTS n they see as 2 Separate squares brackets leave space to fill in the missing number IF 2kt I y Hoyt 25 17 1 25 Mmm we do not add 1110132 1 127 and subtract in this case Now since it is an we can simply add ftp.t.fi to the right whatever we add to the left Www Homework Exercise 1 Ipg192 1 3 1 State whether the following equations represent a circle or not