193 



2. E: 

 M: 

 (2) 



3. E: 

 M: 

 (2) 



4. E: 

 M. 



(2) 



5. E: 

 M: 



x + 3y + z = 0. 

 1 3 



3 3 1 



6 t= + 3 t + 1 = 



X + 2 y + z = 0. 



2 3 



t2 + 2 t + 1 



A = — G, B = 1, C = 3. 

 No real root: no line. 



A = — 1, B = 1, C = 2. 



Only one root: no line 



8x + 3y + z— 1 =0 

 3 3 —3 



3 3 



t + 1 = 



A = 0, B = 18, C = 18 

 Only one root: no line. 



3x — 3y + z — 1 =0 

 3 —3 —3 



A = B = C = 0. 

 3 3 3 



E osculates K at (1, 1, 1) i. e. where t = 3/3 = 1, see §5. 



a) Consider the point P (— 2, 1, 10) on E 



1 2 1 



M': A' = — 3, B' = — 12, C = — 22. 



2 1 —10 



D = 360; two lines on E through P; 

 (3) t' + 6 t2 + 3 t — 10 = ti = 1, t . = — 2, tj = — 5 



Lines through P: 

 Li; X = — 2 — u, y = 1 + u, z = 10 + 6 u 



through which pass osculating planes 



3 X — 3 y + z — 1 = 0, 12 X 4- 6 y + z + 8 = 0. 



L2: X = — 2 — u, y = 1 4- 4 u, z = 10 + 15 u. 



through which pass osculating planes 



3 X — 3 y + z — 1 = 0, 75 X + 15 y + z + 125 = 0. 



b) Consider the point P ( 2, 3, 4) on E 

 1 —2 31 



M': 



(3) 



-2 3 



t^' — 6 t = + 9 t — 

 Line through P: 

 L: X = 2 + u, y = 3 + 5 u, z = 4 + 12 u 



13—33213 



, A' = — 1, B' = 2, C = — 1, D = 0. 

 0, ti = 1, t 2 = 4 



