If [tex]f(x, y, z) = c[/tex] represent a family of surfaces for different values of the constant [tex]c[/tex]. The gradient of the function [tex]f[/tex] defined as [tex]\nabla f[/tex] is a vector normal to the surface [tex]f(x, y, z) = c[/tex].
Given the paraboloid
[tex]y = x^2 + z^2[/tex].
We can rewrite it as a scalar value function f as follows:
[tex]f(x,y,z)=x^2-y+z^2=0[/tex]
The normal to the paraboloid at any point is given by:
[tex]\nabla f= i\frac{\partial}{\partial x}(x^2-y+z^2) - j\frac{\partial}{\partial y}(x^2-y+z^2) + k\frac{\partial}{\partial z}(x^2-y+z^2) \\ \\ =2xi-j+2zk[/tex]
Also, the normal to the given plane [tex]3x + 2y + 7z = 2[/tex] is given by:
[tex]3i+2j+7k[/tex]
Equating the two normal vectors, we have:
[tex]2x=3\Rightarrow x= \frac{3}{2} \\ \\ -1=2 \\ \\ 2z=7\Rightarrow z= \frac{7}{2} [/tex]
Since, -1 = 2 is not possible, therefore there exist no such point on the paraboloid [tex]y = x^2 + z^2[/tex] such that the tangent plane is parallel to the plane 3x + 2y + 7z = 2.
