# Finding the equation of a tangent line

• October 22nd 2013, 04:30 PM
grandmarquis84
Finding the equation of a tangent line
The equation of the paraboloid is given as z=x^2+y^2 at the point x+1 and y=2, and I was wondering how to figure out the equation of the tangent as well as the normal line to the tangent. I know the graph will sort of look like a "bowl" around the z axis, and I know the normal will cross the z-axis as well as another point of intersection on the bowl.

But how do I go about finding the equation of the tangent plane? Thank you in advance.
• October 22nd 2013, 05:17 PM
Prove It
Re: Finding the equation of a tangent line
There will be infinitely many tangents. You need to decide which direction you want your tangent to go in.
• October 22nd 2013, 05:42 PM
HallsofIvy
Re: Finding the equation of a tangent line
Didn't I do this earlier today? I specifically remember the "point x+ 1 and y= 2" and thinking, "Okay, he really means x= 1 and y= 2". In case I only imagined responding that, here's what I imagine I said: The equation $z= x^2+ y^2$ is the same as $x^2+ y^2- z= 0$ so we can this surface as a level surface for $f(x,y,z)= x^2+ y^2- z$. then $\nabla f= 2x\vec{i}+ 2y\vec{j}- \vec{k}$ is perpendicular to the surface at (x, y, z). In particular, at (x, y, z)= (1, 2, 5), that is $2\vec{i}+ 4\vec{j}- \vec{k}$ is the normal vector to the surface so 2(x- 1)+ 4(y- 2)- (z- 5)= 0 is the tangent plane. Of course, the normal line is given by (x, y, z)= (2, 4, -1)t+ (1, 2, 5).