Here's my problem:
Use the normal gradient vector to determine the equation of the line/plane tangent to the given curve/surface at point P.
$x^4 + xy + y^2 = 19$, P(2,-3)
I know how to use the gradient, and end up with an answer of $29x - 4y = ?$. In my book, the given answer is 29x + 4y = 70. Where is this 70 coming from? I'm having a lot of trouble trying to figure it out. Can anyone lend a hand?
The gradient of $f(x,y)=x^4+xy+y^2$ at $P(2,3)$ is $(29,-4)$ and it is perpendicular to the curve $x^4+xy+y^2=19.$ The equation of the tangent line is $$(29,-4)\cdot (x-2,y+3)=0.$$ (Note that if $(x,y)$ is a point of the tangent line then the vector $(x-2,y+3)=(x,y)-(2,-3)$ is perpendicular to the gradient and, thus, its dot product must be zero). That is, the equation of the tangent is
$$29x-4y-70=0.$$