If the curve `C` in the `x y` plane has the equation `x^2+x y+y^2=1,` then the fourth power of the greatest distance of a point on `C` from the origin is___.

Answer

To find the fourth power of the greatest distance of a point on the curve \( C \) defined by the equation \( x^2 + xy + y^2 = 1 \) from the origin, we can follow these steps:

### Step 1: Understand the Distance Formula
The distance \( r \) from the origin to a point \( (x, y) \) is given by:
\[ r = \sqrt{x^2 + y^2} \]
We need to maximize \( r^4 \), which is equivalent to maximizing \( (x^2 + y^2)^2 \).

### Step 2: Express \( y \) in Terms of \( x \)
From the given equation of the curve, we can express \( y \) in terms of \( x \):
\[ y^2 + xy + x^2 = 1 \]
This is a quadratic equation in \( y \):
\[ y^2 + xy + (x^2 - 1) = 0 \]
Using the quadratic formula, we can find \( y \):
\[ y = \frac{-x \pm \sqrt{x^2 - 4(x^2 - 1)}}{2} \]

### Step 3: Substitute \( y \) into the Distance Formula
We will substitute the expression for \( y \) back into the distance formula:
\[ r^2 = x^2 + y^2 \]
Using the quadratic formula, we can find \( y^2 \) and substitute it back into the equation.

### Step 4: Maximize \( r^2 \)
To find the maximum value of \( r^2 \), we can use the method of Lagrange multipliers or substitute \( y \) in terms of \( x \) and maximize the resulting expression. However, we can also use parametric equations to simplify the process.

### Step 5: Use Parametric Representation
Let:
\[ x = r \cos \theta, \quad y = r \sin \theta \]
Substituting these into the curve equation:
\[ (r \cos \theta)^2 + (r \cos \theta)(r \sin \theta) + (r \sin \theta)^2 = 1 \]
This simplifies to:
\[ r^2 (\cos^2 \theta + \sin^2 \theta + \cos \theta \sin \theta) = 1 \]
Using \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[ r^2 (1 + \cos \theta \sin \theta) = 1 \]
Thus:
\[ r^2 = \frac{1}{1 + \cos \theta \sin \theta} \]

### Step 6: Find Maximum \( r^2 \)
To maximize \( r^2 \), we need to minimize \( 1 + \cos \theta \sin \theta \). The maximum value of \( \cos \theta \sin \theta \) is \( \frac{1}{2} \) (which occurs at \( \theta = \frac{\pi}{4} \)):
\[ 1 + \cos \theta \sin \theta \text{ minimum is } 1 + \frac{1}{2} = \frac{3}{2} \]
Thus:
\[ r^2_{\text{max}} = \frac{1}{\frac{3}{2}} = \frac{2}{3} \]

### Step 7: Calculate \( r^4 \)
Now we need to find \( r^4 \):
\[ r^4_{\text{max}} = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \]

### Final Answer
The fourth power of the greatest distance of a point on the curve from the origin is:
\[ \boxed{\frac{4}{9}} \]