`x^2-x y+y^2-4x-4y+16=0` represents a. a point b. a circle c. a pair of straight line d. none of these

Options

A. a point
B. a circle
C. a pair of straight lines
D. none of these

Explanation

To determine what the equation \( x^2 - xy + y^2 - 4x - 4y + 16 = 0 \) represents, we can analyze it step by step.

### Step 1: Rearranging the equation
We start with the given equation:
\[ x^2 - xy + y^2 - 4x - 4y + 16 = 0 \]

### Step 2: Grouping terms
We can rearrange the terms to group the quadratic terms and the linear terms:
\[ x^2 - 4x - xy + y^2 - 4y + 16 = 0 \]

### Step 3: Completing the square
Next, we will complete the square for the \(x\) and \(y\) terms.

For \(x^2 - 4x\):
\[ x^2 - 4x = (x - 2)^2 - 4 \]

For \(y^2 - 4y\):
\[ y^2 - 4y = (y - 2)^2 - 4 \]

Substituting these back into the equation gives:
\[ (x - 2)^2 - 4 - xy + (y - 2)^2 - 4 + 16 = 0 \]
This simplifies to:
\[ (x - 2)^2 + (y - 2)^2 - xy + 8 = 0 \]

### Step 4: Analyzing the equation
Now, we can analyze the equation further. We can isolate the \(xy\) term:
\[ (x - 2)^2 + (y - 2)^2 + 8 = xy \]

### Step 5: Finding the discriminant
To determine the nature of the conic represented by the equation, we can use the discriminant method. The general form of a conic section is given by:
\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]
where \(A = 1\), \(B = -1\), \(C = 1\), \(D = -4\), \(E = -4\), and \(F = 16\).

The discriminant \(D\) is given by:
\[ D = B^2 - 4AC \]
Substituting the values:
\[ D = (-1)^2 - 4(1)(1) = 1 - 4 = -3 \]

### Step 6: Conclusion
Since the discriminant \(D < 0\), this indicates that the equation represents a point (a degenerate conic).

Thus, the answer is:
**a. a point**