If x and y are real numbers connected by the equation `9x ^(2)+2xy+y^(2) -92x-20y+244=0,` then the sum of maximum value of x and the minimum value of y is :

Answer

To solve the problem, we need to find the maximum value of \( x \) and the minimum value of \( y \) from the given quadratic equation:

\[ 9x^2 + 2xy + y^2 - 92x - 20y + 244 = 0 \]

### Step 1: Rewrite the equation in terms of \( x \)

We can treat this equation as a quadratic in \( x \):

\[ 9x^2 + (2y - 92)x + (y^2 - 20y + 244) = 0 \]

### Step 2: Apply the condition for real roots

For \( x \) to have real values, the discriminant \( D \) of the quadratic equation must be non-negative:

\[ D = b^2 - 4ac \geq 0 \]

Here, \( a = 9 \), \( b = 2y - 92 \), and \( c = y^2 - 20y + 244 \). Thus, the discriminant is:

\[ D = (2y - 92)^2 - 4 \cdot 9 \cdot (y^2 - 20y + 244) \geq 0 \]

### Step 3: Simplify the discriminant

Calculating \( D \):

\[ D = (2y - 92)^2 - 36(y^2 - 20y + 244) \]

Expanding \( D \):

\[ D = 4y^2 - 368y + 8464 - 36y^2 + 720y - 8784 \]

Combining like terms:

\[ D = -32y^2 + 352y - 320 \]

### Step 4: Set the discriminant to be non-negative

We need to solve:

\[ -32y^2 + 352y - 320 \geq 0 \]

Dividing through by -16 (and reversing the inequality):

\[ 2y^2 - 22y + 20 \leq 0 \]

### Step 5: Factor the quadratic

Factoring gives:

\[ (y - 10)(y - 1) \leq 0 \]

### Step 6: Determine the intervals

The roots are \( y = 1 \) and \( y = 10 \). The solution to the inequality \( (y - 10)(y - 1) \leq 0 \) gives:

\[ 1 \leq y \leq 10 \]

Thus, the minimum value of \( y \) is \( 1 \).

### Step 7: Solve for maximum \( x \)

Now we will find the maximum value of \( x \) by substituting the values of \( y \) back into the quadratic equation. We will use the maximum value of \( y = 10 \):

Substituting \( y = 10 \):

\[ 9x^2 + (2(10) - 92)x + (10^2 - 20(10) + 244) = 0 \]

This simplifies to:

\[ 9x^2 - 72x + 24 = 0 \]

### Step 8: Calculate the discriminant for \( x \)

The discriminant for this equation is:

\[ D = (-72)^2 - 4 \cdot 9 \cdot 24 = 5184 - 864 = 4320 \]

### Step 9: Find the roots

Using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{72 \pm \sqrt{4320}}{18} \]

Calculating \( \sqrt{4320} \approx 65.7 \):

\[ x = \frac{72 \pm 65.7}{18} \]

Calculating the two possible values:

1. \( x = \frac{137.7}{18} \approx 7.65 \)
2. \( x = \frac{6.3}{18} \approx 0.35 \)

Thus, the maximum value of \( x \) is approximately \( 7.65 \).

### Step 10: Sum the maximum \( x \) and minimum \( y \)

Finally, we find the sum:

\[ \text{Sum} = \text{Maximum } x + \text{Minimum } y = 7.65 + 1 = 8.65 \]

### Conclusion

The sum of the maximum value of \( x \) and the minimum value of \( y \) is approximately \( 8.65 \).

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