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Find the point on the surface 6x = y2 + z2 so that its tangent plane is parallel to 3x + 2y - 2z = 1.

Answer 1

# Given Surface : 6x=9 % + 22 patrallel to 3x+2y-22-8 Sell Given a surface defined implicitly by F(X, Y, 2) =0 the Egn of the tangent plane to the surface face at a is given by paint (a, b, c) DF (a, b, c). (7-9, 9-6, 2-c) = 0 or equivalently Fala,b,c)(x-9) + Fy (9,6,0)(-b)+ Fz(0,6,c)(2-0) 20 It is important to remember that IF (a, b, () is a normal vector to the tangent plane at (a, b, c) Let's rewrite the equation of the parabolaid as 6x-42 - 2²²0 and Let F(x, y, 2) = 6x-42z² Note Ex(x, y, z) = 6 Fy(x,y 2)=-24 . Fz (X,Y, 2) =-22 We wants the point (20, 96, 20) at which the tangent palane is parallel to the plane 3x+2y-22-1=0 which has normal vector <3, 2,-2) The planes are parallel when they have parallel normal vectors i.e when the normal vector of our tangent plane is multiple at <3,2,-2) Thus <6,-2% -220) = C{3,2,-2) 6=30 ,- 240=26, -22. =-20 2.=2 Yo = -2 - c=2 Zo=C

So the point where the tangent plane is parallel to The plane 3x+2y-22-1-0 is at (6,-2, 2) (No, Yo, Zo) (6.-2, 2) Ans