The graph of the curve `x^2+y^2-2x y-8x-8y+32=0` falls wholly in the (a) first quadrant (b) second quadrant third quadrant (d) none of these

Options

A. first quadrant
B. second quadrant
C. third quadrant
D. none of these

Explanation

To determine the quadrant in which the graph of the curve \(x^2 + y^2 - 2xy - 8x - 8y + 32 = 0\) falls wholly, we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ x^2 + y^2 - 2xy - 8x - 8y + 32 = 0 \]
We can rearrange the terms to group them conveniently:
\[ x^2 - 2xy + y^2 - 8x - 8y + 32 = 0 \]

### Step 2: Complete the square
Notice that \(x^2 - 2xy + y^2\) can be rewritten as \((x - y)^2\). Thus, we have:
\[ (x - y)^2 - 8x - 8y + 32 = 0 \]
Next, we will complete the square for the terms involving \(x\) and \(y\).

### Step 3: Rearranging the equation
We can rewrite the equation as:
\[ (x - y)^2 = 8x + 8y - 32 \]
Now, let's rearrange the right-hand side:
\[ (x - y)^2 = 8(x + y - 4) \]

### Step 4: Analyze the equation
This equation suggests that the curve is a parabola. The vertex of the parabola can be derived from the equation \(x + y - 4 = 0\), which is a line. The vertex will be the point where this line intersects the axis.

### Step 5: Find the intercepts
To check where the curve intersects the axes, we can set \(x = 0\) and \(y = 0\) separately.

1. **Finding y-intercepts (set \(x = 0\)):**
\[ 0^2 + y^2 - 2(0)y - 8(0) - 8y + 32 = 0 \]
Simplifying gives:
\[ y^2 - 8y + 32 = 0 \]
The discriminant \(D = b^2 - 4ac = (-8)^2 - 4(1)(32) = 64 - 128 = -64\) (less than 0). Thus, there are no real roots, meaning the curve does not intersect the y-axis.

2. **Finding x-intercepts (set \(y = 0\)):**
\[ x^2 + 0^2 - 2x(0) - 8x - 8(0) + 32 = 0 \]
Simplifying gives:
\[ x^2 - 8x + 32 = 0 \]
The discriminant \(D = (-8)^2 - 4(1)(32) = 64 - 128 = -64\) (also less than 0). Thus, there are no real roots, meaning the curve does not intersect the x-axis either.

### Step 6: Determine the quadrant
Since the curve does not intersect either axis, we need to analyze the behavior of the curve. The vertex derived from the line \(x + y - 4 = 0\) indicates that the curve opens upwards or downwards. Given that both discriminants are negative, the curve must lie entirely above the line \(x + y = 4\).

### Conclusion
Since the curve does not intersect the axes and lies above the line \(x + y = 4\), it must fall entirely in the **first quadrant**.

### Final Answer
The graph of the curve falls wholly in the **(a) first quadrant**.