The equation of the major axis of the ellipse `(x-1)^(2)/(9)+(y-6)^(2)/(4)=1` is

Options

A. y-2=0
B. x-1=0
C. y=6
D. x=5

Answer

To find the equation of the major axis of the given ellipse \((x-1)^{2}/9 + (y-6)^{2}/4 = 1\), we can follow these steps:

### Step 1: Identify the standard form of the ellipse
The standard form of the equation of an ellipse is given by:
\[ \frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1 \]
where \((h, k)\) is the center of the ellipse, \(a\) is the semi-major axis, and \(b\) is the semi-minor axis.

### Step 2: Compare with the given equation
The given equation is:
\[ \frac{(x-1)^{2}}{9} + \frac{(y-6)^{2}}{4} = 1 \]
From this, we can identify:
- \(h = 1\)
- \(k = 6\)
- \(a^{2} = 9\) (thus \(a = 3\))
- \(b^{2} = 4\) (thus \(b = 2\))

### Step 3: Determine the orientation of the ellipse
Since \(a > b\) (3 > 2), this indicates that the major axis is vertical.

### Step 4: Write the equation of the major axis
For an ellipse where the major axis is vertical, the equation of the major axis is given by:
\[ x = h \]
Substituting the value of \(h\):
\[ x = 1 \]

### Step 5: Identify the equation of the minor axis
The equation of the minor axis is given by:
\[ y = k \]
Substituting the value of \(k\):
\[ y = 6 \]

### Conclusion
The equation of the major axis of the ellipse is:
\[ x = 1 \]