Question

Find the points on the hyperboloid x^2+4y^2-z^2=4 where the tangent plane is parallel to the plane 2x+2y+z=5

Answer 1

To find the points on the hyperboloid x^2 + 4y^2 - z^2 = 4 where the tangent plane is parallel to the plane 2x + 2y + z = 5, calculate the gradient of the hyperboloid to get the normal vectors, then set up a system of equations involving the scalar multiple relating these normal vectors.

Step-by-step explanation:

The question involves finding the points on a hyperboloid defined by the equation x^2 + 4y^2 - z^2 = 4 where the tangent plane is parallel to a given plane with equation 2x + 2y + z = 5. To solve this, one must understand that two planes are parallel when their normal vectors are scalar multiples of each other. So, we need to find the gradient of the hyperboloid function which gives us the normal vector at any point on the hyperboloid. The gradient is calculated by taking the partial derivatives of the function with respect to x, y, and z, giving us (2x, 8y, -2z) as the normal vector.

Since we are looking for it to be parallel to the plane with the normal vector (2, 2, 1), we can set up a system of equations to find the scalar multiple that relates both vectors. We are looking for points where 2x = 2k, 8y = 2k, and -2z = k for some scalar k, and these points must also satisfy the hyperboloid equation. We end up with a system of equations that can be solved simultaneously to find the required points on the hyperboloid.

Answer 2

Finding points on a hyperboloid where the tangent plane is parallel to a given plane involves using the gradient of the hyperboloid's equation to determine the normal vector and equating it to the normal vector of the given plane, up to a scalar multiple.

Step-by-step explanation:

To find the points on the hyperboloid x^2 + 4y^2 - z^2 = 4 where the tangent plane is parallel to the plane 2x + 2y + z = 5, we need to find the normal vector of the given hyperboloid's surface. The normal vector can be found using the gradient of the hyperboloid's equation, which gives us the coefficients of the x, y, and z terms in the tangent plane's equation. In this case, we take the partial derivatives of the hyperboloid's equation with respect to x, y, and z to obtain the normal vector (2x, 8y, -2z).

Since the tangent plane must be parallel to 2x + 2y + z = 5, it must have the same normal vector except for a scalar multiple. The normal vector of the plane 2x + 2y + z is (2, 2, 1). We can set the hyperboloid's normal vector equal to this plane's normal vector, up to a scalar multiple, to find the points (x, y, z) that satisfy this condition.