At what point on the paraboloid $\displaystyle y=x^2+z^2$ is the tangent plan parallel to the plane $\displaystyle x+2y+3z = 1$.
I just need some help getting started.
A normal to the given plane is i + 2j + 3k.
A normal to the paraboloid is $\displaystyle \frac{\partial y}{\partial x} i - j + \frac{\partial y}{\partial z} k = (2x) i - j + (2z) k$.
A normal of the tangent plane at the point (a, a^2 + b^2, b) is therefore (2a) i - j + (2b) k.
You want to find the point at which the tangent plane has a normal parallel to i + 2j + 3k. You therefore want the values of a and b such that $\displaystyle (2a) i - j + (2b) k = \mu (i + 2j + 3k)$ where $\displaystyle \mu$ is a scalar. So solve the following equations simultaneously:
$\displaystyle 2a = \mu$ ... (1)
$\displaystyle -1 = 2 \mu$ ... (2)
$\displaystyle 2b = 3 \mu$ ... (3)