The coentre of the ellipse `4x ^(2) + 9y ^(2) -16x-54y+61=0` is

Options

A. `(1,3)`
B. `(2,3)`
C. `(3,2)`
D. `(3,1)`

Answer

To find the center of the ellipse given by the equation \(4x^2 + 9y^2 - 16x - 54y + 61 = 0\), we will follow these steps:

### Step 1: Rearrange the equation
We start with the equation:
\[ 4x^2 + 9y^2 - 16x - 54y + 61 = 0 \]
We can rearrange it by moving the constant term to the other side:
\[ 4x^2 + 9y^2 - 16x - 54y = -61 \]

### Step 2: Group the x and y terms
Next, we group the \(x\) terms and the \(y\) terms:
\[ 4(x^2 - 4x) + 9(y^2 - 6y) = -61 \]

### Step 3: Complete the square for x
To complete the square for the \(x\) terms, we take the coefficient of \(x\), which is \(-4\), halve it to get \(-2\), and square it to get \(4\):
\[ 4(x^2 - 4x + 4 - 4) = 4((x - 2)^2 - 4) = 4(x - 2)^2 - 16 \]

### Step 4: Complete the square for y
Now, we complete the square for the \(y\) terms. The coefficient of \(y\) is \(-6\), halve it to get \(-3\), and square it to get \(9\):
\[ 9(y^2 - 6y + 9 - 9) = 9((y - 3)^2 - 9) = 9(y - 3)^2 - 81 \]

### Step 5: Substitute back into the equation
Substituting back into the equation, we have:
\[ 4((x - 2)^2 - 4) + 9((y - 3)^2 - 9) = -61 \]
This simplifies to:
\[ 4(x - 2)^2 - 16 + 9(y - 3)^2 - 81 = -61 \]
Combining the constants gives:
\[ 4(x - 2)^2 + 9(y - 3)^2 - 97 = -61 \]
Adding \(97\) to both sides results in:
\[ 4(x - 2)^2 + 9(y - 3)^2 = 36 \]

### Step 6: Divide by 36
Now, divide the entire equation by \(36\):
\[ \frac{4(x - 2)^2}{36} + \frac{9(y - 3)^2}{36} = 1 \]
This simplifies to:
\[ \frac{(x - 2)^2}{9} + \frac{(y - 3)^2}{4} = 1 \]

### Step 7: Identify the center
From the standard form of the ellipse \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\), we can identify the center \((h, k)\):
\[ h = 2, \quad k = 3 \]
Thus, the center of the ellipse is:
\[ (2, 3) \]

### Final Answer
The center of the ellipse is \((2, 3)\).