Find the equation of tangent line to the intersection of the two surfaces in $P(4,-2,20)$
$$z=x^2+y^2 \,\,,\,z=2x+4y+20$$
if question like this is given ,i can easily equate both equations ,find gradient and compute the tangent line passing through the point .
But what if surfaces are given as P(x,y,z)=c and Q(x,y,z)=d,with P(x,y,z) and Q(x,y,z) both differentiable how should i find equation of the tangent line passing through a point $(x_0,y_0,z_0)$
Best Answer
Short answer is that a vector parallel to the line is $v=\nabla P \times \nabla Q.$
Since $\nabla P$ is perpendicular to the first surface, $v$ is parallel to it. Since $\nabla Q$ is perpendicular to the second surface, $v$ is parallel to that. So $v$ is parallel to both surfaces, so it must be tangent to the curve.