To find a point on the hyperboloid $x^2 + 4y^2 - z^2 = 1$ where the tangent plane is parallel to the plane $x + 4y - z = 0$, follow these steps:

Step 1: Gradient of the Hyperboloid

The equation of the hyperboloid is: $f(x, y, z) = x^2 + 4y^2 - z^2 - 1 = 0$ The normal vector to the surface at any point is given by the gradient of the function: $\nabla f(x, y, z) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$ Calculating the partial derivatives: $\frac{\partial f}{\partial x} = 2x, \quad \frac{\partial f}{\partial y} = 8y, \quad \frac{\partial f}{\partial z} = -2z$ Thus, the gradient is: $\nabla f(x, y, z) = (2x, 8y, -2z)$ This vector is normal to the tangent plane at any point on the hyperboloid.

Step 2: Normal Vector to the Given Plane

The equation of the given plane is $x + 4y - z = 0$. The normal vector to this plane is simply the coefficients of $x$, $y$, and $z$: $\mathbf{n} = (1, 4, -1)$

Step 3: Tangent Plane Condition

For the tangent plane to the hyperboloid to be parallel to the given plane, their normal vectors must be proportional. Therefore, the gradient $(2x, 8y, -2z)$ should be a scalar multiple of $(1, 4, -1)$. This gives the system of equations: $2x = \lambda \cdot 1, \quad 8y = \lambda \cdot 4, \quad -2z = \lambda \cdot (-1)$ Simplifying these equations: $2x = \lambda, \quad 8y = 4\lambda, \quad -2z = \lambda$ From the first and third equations, we get: $x = \frac{\lambda}{2}, \quad z = \frac{\lambda}{2}$ From the second equation: $8y = 4\lambda \quad \Rightarrow \quad y = \frac{\lambda}{2}$

Step 4: Substituting into the Hyperboloid Equation

Now substitute $x = \frac{\lambda}{2}, y = \frac{\lambda}{2}, z = \frac{\lambda}{2}$ into the hyperboloid equation $x^2 + 4y^2 - z^2 = 1$: $\left( \frac{\lambda}{2} \right)^2 + 4\left( \frac{\lambda}{2} \right)^2 - \left( \frac{\lambda}{2} \right)^2 = 1$ Simplifying: $\frac{\lambda^2}{4} + 4 \cdot \frac{\lambda^2}{4} - \frac{\lambda^2}{4} = 1$ [ \frac{\lambda^2}{4} + \lambda^2 - \frac{\lambda