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.
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.
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.
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.
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.