Question

Find an equation of the tangent plane and parametric equations of the normal line to the surface ?? − ?? 3 + ?? 2 = 2 at the point ?(−1, −1, 2).

Answer 1

Given Surface az - yz² + y z = 2 (= F say) given point Nova P (-1,-1, 2) 21, Y, Z, Tangent plane to the surface F at the point P is given by the formula OF (X-X1) + (Y-y, at P af OF dx at P + 12-2,) by at P sz (x-(I)) (2) + (y-(-1)) 6-4) + (2-2) (7) OF N 11 . әх = 2 at -2² +22; ax at P Hols OF -8+4 = -4 y at P ul x-3y2 + 2yz; af -1-3(-1)(2) + 2 2(+1) (2) az JZ atp = -1 +12 -4 = 12-5-7 S=7 => 2 (x + 1) -414+1) +7(2-2) = 0 2x + 2 -44 - 4 + 77-14-0 0 1 2X-44 +72-16 = 0.1 The normal live to f at point (81,4,,?,) is X(t) = Witt of Txat P = - + t2 = 27-1 ylt) = y, t t of = -1+7(4) = -4t-1 ay at P z(t) = 34+ E OF IZ at P 2+t(7) - 7t+2 o'. Parametric Equation of the nomal line is X(t)= 2t-] ; y(t) = -44-1, 2(t) = 75+2