Question

4. [20 marks] (a) Write down in standard form and then identify the quadric surface 2.2 - 3.02 - 2 = 4y? (b) Find the center and radius of the sphere 3.c2 + 3y2 + 3:2 + 18.6 - 12y + 362 = 0. (c) Find an arc length parametrization of the curve r(t) = {{i + £taj, t > 0. un df aw dr (d) Show that I r(t)) = (t) + f(t) , where r(t) = x(t)i + y(t)j+z(t)k is a vector function, and f(t), c(t), y(t) and z(t) are scalar functions.

Answer 1

(a) 22² -3 11 ² - 2 = 4 4 2 or - 3 012 - 4 y² +22² =2 2 - 1 + 2 2 = 1 02-32 32 This is hyferbaloid of twce sheets with 2-a48 as its aris (6) Criven sphory is 12 mdar 384 +37732 +18X-12y+36250 on x + 9 +2 ² + 6X-Q4 +127-0 Comparing with x² + y +22touxte og tawatd=0 aga U-3 U = -2, W = 6 d.o. Centre of 8 there is (-4,-0-W) as (-3, 2-6) cs Scanned with CamScanner

Radius = 42 +62+w2d - 9+ 4 +36-0= 7 (0) 37 = 2 + + 12 of too Here = 1 + y = 1 & 2 SE See +672 + Haar 14 +++ cs samt at P24142 - $* Sc CamScanner

Putting 1² 4 an t dt = f dy 14 u 2412 - dag 1 16 out THR a S.Com ** SHP - If Con}_cuauitate A 12 -- (244) 2.2cm MAJH2-EC 100-2016 -(-2)] ****- [{ coa)-recent COLD cs Scanned with CamScanner

(d) 361) = x + 4A) +26+) - CFO) a) Tía) (xay xay 420) ) fel). + fc13014 fayzechy le ] (fa) x4) J+55 cfayeb) te [f() 264)] erotar 18 Eî [ 8.CO) x64 +fc) dxa) 245 ellery a). + ftu) ty + i defe y co ) + f(x) $999. - effet cxu thy (4) +11 216) It fafi dict) +5 auto + d 2. = d.ge 34) + fa) dilan) cs separed +) Ž(1)) = 742 + fw duz dt 0+ 3+ Scanned with CamScanner