Find a Tangent Plane Parallel to x+2y+3z=1 on the Curve y=x2+z2

In summary, the question asks for the point on the parabolic surface y=x^2+z^2 that has a tangent plane parallel to the plane x+2y+3z=1. To solve this, you need to find the gradient of the function y=x^2+z^2 and use it to find the normal vector of the tangent plane. From there, you can use the known point and the normal vector to find the equation of the tangent plane.
  • #1
TheColorCute
22
0
I'm completely lost on this question, and it's due tomorrow morning. Help?

Homework Statement


What point on y=x2+z2 is the tangent plane parallel to the plane x+2y+3z=1?

Homework Equations


y=x2+z2
x+2y+3z=1

The Attempt at a Solution


I have no idea what to do...

Thanks!
 
Last edited:
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  • #2
The gradient of the function F(x,y,z) is always perpendicular to the surface F(x,y,z)= constant at the point (x,y,z). Now, do you know how to find the equation of a plane, given a normal vector and one point on that plane?
 

FAQ: Find a Tangent Plane Parallel to x+2y+3z=1 on the Curve y=x2+z2

What is a tangent plane?

A tangent plane is a plane that touches a curve or surface at exactly one point. It is perpendicular to the curve or surface at that point.

How do you find the equation of a tangent plane?

To find the equation of a tangent plane, you need the coordinates of the point where the plane touches the curve or surface, as well as the slope or gradient of the curve or surface at that point. Using this information, you can use the formula for a plane (Ax + By + Cz = D) to find the values of A, B, C, and D for the tangent plane.

Can you find a tangent plane parallel to a given plane?

Yes, it is possible to find a tangent plane parallel to a given plane. This can be done by finding a point on the given plane and using the slope or gradient of the plane to find the equation of the tangent plane at that point.

What is the equation of the curve in this problem?

The equation of the curve in this problem is y = x2 + z2. This is a 3D parabolic curve that extends infinitely in both the x and z directions.

How do you solve this problem?

To solve this problem, you need to find a point on the curve y = x2 + z2 that also lies on the given plane x + 2y + 3z = 1. This can be done by substituting x2 + z2 for y in the plane equation, resulting in a quadratic equation in terms of x and z. Solving this equation will give you the coordinates of the point on the curve. From there, you can use the slope of the curve at that point to find the equation of the tangent plane using the formula for a plane.

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