the centre of the ellipse `((x+y-2)^(2))/(9)+((x-y)^(2))/(16)=1` , is

Options

A. (0,0)
B. (1,0)
C. (0,1)
D. (1,1)

Answer

To find the center of the ellipse given by the equation

\[ \frac{(x+y-2)^2}{9} + \frac{(x-y)^2}{16} = 1, \]

we can follow these steps:

### Step 1: Identify the general form of the ellipse
The given equation is already in the standard form of an ellipse, where we can identify the expressions inside the squares.

### Step 2: Rewrite the equation
The equation can be rewritten as:

\[ (x+y-2)^2 = 9(1 - \frac{(x-y)^2}{16}). \]

### Step 3: Find the center
The center of the ellipse can be found by determining the values of \(x\) and \(y\) that make both expressions equal to zero.

1. Set \(x + y - 2 = 0\) (from the first term).
2. Set \(x - y = 0\) (from the second term).

### Step 4: Solve the equations
From the first equation:

\[ x + y = 2 \quad \text{(1)} \]

From the second equation:

\[ x = y \quad \text{(2)} \]

Substituting equation (2) into equation (1):

\[ x + x = 2 \implies 2x = 2 \implies x = 1. \]

Now substituting \(x = 1\) back into equation (2):

\[ y = 1. \]

### Step 5: Conclusion
Thus, the center of the ellipse is at the point \((1, 1)\).

### Final Answer
The center of the ellipse is \((1, 1)\).

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