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To find the equation of the tangent plane to the given surface at the specified point, let’s go through the following steps.

### Problem Analysis

1. **Surface Equation**:
   \[
   x^2 + \frac{y^2}{9} + \frac{z^2}{9} = 1
   \]
   This is an ellipsoid.

2. **Point of Tangency**:
   The given point is \(\left( \frac{1}{3}, 2, 2 \right)\).

3. **Equation of the Tangent Plane**:
   For a surface defined implicitly by \(F(x, y, z) = 0\), the equation of the tangent plane at a point \((x_0, y_0, z_0)\) is given by:
   \[
   F_x(x_0, y_0, z_0)(x - x_0) + F_y(x_0, y_0, z_0)(y - y_0) + F_z(x_0, y_0, z_0)(z - z_0) = 0
   \]
   where \(F_x\), \(F_y\), and \(F_z\) are the partial derivatives of \(F\) with respect to \(x\), \(y\), and \(z\), respectively.

### Step 1: Define \(F(x, y, z)\)
Let
\[
F(x, y, z) = x^2 + \frac{y^2}{9} + \frac{z^2}{9} - 1
\]
so that \(F(x, y, z) = 0\) describes our surface.

### Step 2: Compute Partial Derivatives
1. \(F_x = \frac{\partial F}{\partial x} = 2x\)
2. \(F_y = \frac{\partial F}{\partial y} = \frac{2y}{9}\)
3. \(F_z = \frac{\partial F}{\partial z} = \frac{2z}{9}\)

### Step 3: Evaluate Partial Derivatives at \(\left( \frac{1}{3}, 2, 2 \right)\)
1. \(F_x\left(\frac{1}{3}, 2, 2\right) = 2 \cdot \frac{1}{3} = \frac{2}{3}\)
2. \(F_y\left(\frac{1}{3}, 2, 2\right) = \frac{2 \cdot 2}{9} = \frac{4}{9}\)
3. \(F_z\left(\frac{1}{3}, 2, 2\right) = \frac{2 \cdot 2}{9} = \frac{4}{9}\)

### Step 4: Substitute into Tangent Plane Formula
Using the formula:
\[
F_x(x_0, y_0, z_0)(x - x_0) + F_y(x_0, y_0, z_0)(y - y_0) + F_z(x_0, y_0, z_0)(z - z_0) = 0
\]
we get:
\[
\frac{2}{3} \left( x - \frac{1}{3} \right) + \frac{4}{9} (y - 2) + \frac{4}{9} (z - 2) = 0
\]

### Step 5: Simplify
Multiply through by 9 to clear the fractions:
\[
6 \left( x - \frac{1}{3} \right) + 4 (y - 2) + 4 (z - 2) = 0
\]
Expanding each term:
\[
6x - 2 + 4y - 8 + 4z - 8 = 0
\]
Combine terms:
\[
6x + 4y + 4z = 18
\]
or
\[
z = -\frac{1}{2}(x - \frac{1}{

Consider the surface x^2 + y^2/9 + z^2/9 = 1. Find the equation of the tangent plane to this surface at the point (1/3, 2, 2).

Learn how to find the tangent plane equation for the surface x^2 + y^2/9 + z^2/9 = 1 at the point (1/3, 2, 2). This solution involves calculus concepts such as partial derivatives and implicit differentiation, suitable for undergraduate calculus students.

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