Question

Question 2: [20 marks Find and plot the Cartesian equations of the loci of z given by the following equations (for a circle indicate explicitly coordinates of its centre and radius; for an ellipse indicate explicitly coordinates of its foci and semi-axes; for a straight line indicate explicitly its gradient and y- intercept, for a ray indicate explicitly coordinates of its origin and gradient): [7 marks) b) arg(2-4+j)= arg(jV3 –1). 16 marks) c) (32 – 21 – 12j|+|3z +9-12j1 = 48. 17 marks)

Answer 1

1) 17+3! == 16j-221 3 Let a=x+jy then 2+3=xtjyt3 2+3 = f (x+3) + jy 64-22= 6i-2(x+3y) = Gj-2x-day (6u-22) = -2x+j (24+6) 312431 = 163-221 Upan stunning we get 9 12 +312 = 165-2212 9[t2+3++y2] = [ 62254+ (-2Y+634] 9[z+y2+62+9] = { @27+ 492736 -244] qa++99454X +81 = 4224497247 +36 527+592 +542 +244 +45=0 alt yet + sen + acqay +9=0 2²+ 2.2ta +(2+2 - (27 2 + y 2.124+ (12) ²017 1920 (2487)++ (84 ta je = (2+++(352_9 = 729 +144–025 = 648 fakt F3++ (5448 y() The laws is the Ciscle with care (27, 12), cs Scanned with ad sedius va ave = 1812 CamScanner 2

Ghen Greg (1-4ts) = Arg (141) Ag (-14 iP) = - tail (17) = . = 29 → AG (2-(4-1)) = 24 2-4+3 = 2417-4+1 = 12-4143 (4ti) Arg (7-4+j) = taül (90) = 21 y = tail 941==3(-a), The ea of the leas is Itla-tentar 2-440 il <4 y=-3x+ ab- with Gel>o ie 371 Gradent is ma-to y-intercept is ak- 137-21-1231 + (329-1231 248 3 12-7-431 +31243-411 - 48 12-(H ) + 12+ 3-uil = 16 12-(taj) + 12-6+43)1 = 16 la tj4(+25)| + 12+15+3 -431 =16 trz y ka-A+44-41| +1043) +(4-4) 1 = 16 laces Jaff+(5-47 +√9+387(9-47² = 16 & Wra-F?+ 1f-a7² = 16 -√2+37+19-47 Opensavaning butulides (a-784 = 286 +6+33_44-– 2:46 Jees trobest CS Scamarded fruit1_04332–266 = -32Jta+35++8472 Cam Scanner

(2-7)+Q+37] [a--3] –276 = -32 7+3+ 2012 (22-4) (-10) - 256 = -32 √ (2+3) ²48-412 -2012-2) – 256 - = -32 √2+3²+ forrit 5(2-2) +64 = 814433448-07 (82–10+64) = 8√2+37219-912 (52+54) = 64(+3)+7 (947) 25274 2916+ 5402 = 66(22+60+9 48371684] 392 +6692-5402-5124 + 3842 - 1316 0 3924-1563 +6442-slay –1316 20 39622_42+4-6] + 64[y?_84+16+63—1316 26 39 {(x-2)²_43 +64 {(8-4)²_16] -1316=0 394-212_156+649-47²_1024_1316 to 39 (22-232 +6469-44 –4496 =6 centre of the ellipee is R24) =Chik) eccentricity es veuen vi v o - Coordinates of the fuci ae Chiae, k) = (2489,4) - (215,4) =64),63,9) The coordinates of fociare ACFG) B(-34) CS Camise en of eemi anes ase 9=4 and 1=2

-5

The above graph is the plot of the Circle equation got for question-1

The above graph is the plot of the locus (half Line) got for the second question

- 10 -8

-10 -8

The above graph is the plot of the locus got for Question-3