# Thread: Equation of Line Tangent to Intersection of Surfaces

1. ## (PLEASE HELP) Equation of Line Tangent to Intersection of z = arctan(xy) and x = 2

Can someone help me with this problem? I'm trying to find the equation of the line tangent to the intersection of the surface z = arctan(xy) with the plane x = 2 at the point (2, 1/2, pi/4).

Since x is held constant we are finding the partial derivative wrt to y, i.e.
d(arctan(xy))/dy = 1/(1+(xy)^2) * x = x/(1+(xy)^2) where d/dy is a partial derivative. The slope of the line is given by the value of (partials) dz/dy at (2,1/2) which is precisely 1. I know now that the parametric equations will be given by:

x = 2
y = 1/2+ ?
z = pi/4 + ?

Can somebody explain to me how to get the terms containing t in the parameterizations for y(t) and z(t)? I would really appreciate it.

2. Originally Posted by sdh2106
Can someone help me with this problem? I'm trying to find the equation of the line tangent to the intersection of the surface z = arctan(xy) with the plane x = 2 at the point (2, 1/2, pi/4).

Since x is held constant we are finding the partial derivative wrt to y, i.e.
d(arctan(xy))/dy = 1/(1+(xy)^2) * x = x/(1+(xy)^2) where d/dy is a partial derivative. The slope of the line is given by the value of (partials) dz/dy at (2,1/2) which is precisely 1. I know now that the parametric equations will be given by:

x = 2
y = 1/2+ ?
z = pi/4 + ?

Can somebody explain to me how to get the terms containing t in the parameterizations for y(t) and z(t)? I would really appreciate it.
You have this intersection's equation $z = \arctan \left(2y\right)$

The general equation of the tangent line (on the yz-plane) has the form ${\color{red}\boxed{\color{black}{z - z_0 = {{z'}_0}\left(y - y_0\right)}}}$

In your case $y_0 = \frac{1}{2},{\text{ }}{z_0} = \frac{\pi }{4}$

Find ${z'}_0$

$z' = \frac{dz}{dy}\arctan \left(2y\right) = \frac{2}{1 + 4y^2} \quad \Rightarrow \quad {{z'}_0} = \frac{2}{1 + 4 \left(1/2\right)^2} = \frac{2}{2} = 1.$

Finally you have $z = y + \frac{\pi }{4} - \frac{1}{2} = y + \frac{\pi -2}{4}.$

3. I appreciate the help, but I was trying to find out how to parameterize this information. If you see my original question, all I am trying to do is find the terms that contain t in y(t) and z(t).