The equation of the major axis of `25x^(2) + 16y^(2)-100x - 96y - 156 = 0` is

Options

A. x-4=0
B. x-2=0
C. y-3=0
D. y-6=0

Answer

To find the equation of the major axis of the given ellipse \(25x^2 + 16y^2 - 100x - 96y - 156 = 0\), we will follow these steps:

### Step 1: Rearrange the equation
Start by rearranging the equation to group the \(x\) and \(y\) terms together:
\[ 25x^2 - 100x + 16y^2 - 96y - 156 = 0 \]

### Step 2: Complete the square for \(x\) terms
Factor out the coefficient of \(x^2\) from the \(x\) terms:
\[ 25(x^2 - 4x) + 16y^2 - 96y - 156 = 0 \]
Now, complete the square for \(x^2 - 4x\):
\[ x^2 - 4x = (x - 2)^2 - 4 \]
So, substituting this back, we get:
\[ 25((x - 2)^2 - 4) + 16y^2 - 96y - 156 = 0 \]
This simplifies to:
\[ 25(x - 2)^2 - 100 + 16y^2 - 96y - 156 = 0 \]

### Step 3: Complete the square for \(y\) terms
Now, factor out the coefficient of \(y^2\) from the \(y\) terms:
\[ 25(x - 2)^2 + 16(y^2 - 6y) - 256 = 0 \]
Complete the square for \(y^2 - 6y\):
\[ y^2 - 6y = (y - 3)^2 - 9 \]
Substituting this back, we have:
\[ 25(x - 2)^2 + 16((y - 3)^2 - 9) - 256 = 0 \]
This simplifies to:
\[ 25(x - 2)^2 + 16(y - 3)^2 - 144 = 0 \]

### Step 4: Rearranging the equation
Rearranging gives:
\[ 25(x - 2)^2 + 16(y - 3)^2 = 144 \]

### Step 5: Divide by 144
Now, divide the entire equation by 144 to get it into standard form:
\[ \frac{25(x - 2)^2}{144} + \frac{16(y - 3)^2}{144} = 1 \]
This simplifies to:
\[ \frac{(x - 2)^2}{\frac{144}{25}} + \frac{(y - 3)^2}{\frac{144}{16}} = 1 \]
Which can be written as:
\[ \frac{(x - 2)^2}{5.76} + \frac{(y - 3)^2}{9} = 1 \]

### Step 6: Identify the major and minor axes
From the standard form \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\), we see:
- \(a^2 = 5.76\) (which means \(a = \sqrt{5.76} = 2.4\))
- \(b^2 = 9\) (which means \(b = 3\))

Since \(b > a\), the major axis is vertical.

### Step 7: Write the equation of the major axis
The center of the ellipse is at \((h, k) = (2, 3)\). The equation of the major axis, which is vertical, is given by:
\[ x = h = 2 \]

Thus, the equation of the major axis is:
\[ \boxed{x = 2} \]