If curve is represneted by `C=21x^(2)-6xy+29y^(2)-58y-151=0` then
The equation of major axis is

Options

A. `x+3y+3=0`
B. `x+3y-3=0`
C. `x-3y+3=0`
D. none of these

Answer

To find the equation of the major axis of the given curve represented by \( C = 21x^2 - 6xy + 29y^2 - 58y - 151 = 0 \), we will follow these steps:

### Step 1: Identify the type of conic section
The given equation is a quadratic in \(x\) and \(y\). To determine the type of conic, we need to calculate the discriminant \(D\) using the formula:
\[ D = B^2 - 4AC \]
where \(A = 21\), \(B = -6\), and \(C = 29\).

### Step 2: Calculate the discriminant
Substituting the values:
\[ D = (-6)^2 - 4 \cdot 21 \cdot 29 \]
\[ D = 36 - 2436 = -2400 \]
Since \(D < 0\), the conic section is an ellipse.

### Step 3: Rewrite the equation in standard form
To find the axes of the ellipse, we need to rewrite the equation in the standard form of an ellipse:
\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]
We will first complete the square for the \(x\) and \(y\) terms in the equation.

### Step 4: Complete the square
Rearranging the equation:
\[ 21x^2 - 6xy + 29y^2 - 58y = 151 \]
We can group the \(x\) and \(y\) terms:
\[ 21x^2 - 6xy + 29(y^2 - 2y) = 151 \]
Completing the square for \(y\):
\[ y^2 - 2y = (y-1)^2 - 1 \]
Substituting back:
\[ 21x^2 - 6xy + 29((y-1)^2 - 1) = 151 \]
This simplifies to:
\[ 21x^2 - 6xy + 29(y-1)^2 - 29 = 151 \]
\[ 21x^2 - 6xy + 29(y-1)^2 = 180 \]

### Step 5: Divide by 180
Now we divide the entire equation by 180 to get it into standard form:
\[ \frac{21x^2 - 6xy + 29(y-1)^2}{180} = 1 \]

### Step 6: Identify the axes
To find the major and minor axes, we need to identify the coefficients of \(x^2\) and \(y^2\) after rewriting the equation. The major axis is determined by the larger denominator in the standard form.

### Step 7: Determine the equation of the major axis
From the standard form, we can determine the direction of the major axis. If \(a^2 > b^2\), the major axis is along the x-axis, and if \(b^2 > a^2\), the major axis is along the y-axis.

In our case, we need to analyze the coefficients after completing the square and rewriting the equation.

After simplifying, we find that the major axis is vertical, leading to the equation:
\[ x = 0 \]

### Final Answer
The equation of the major axis is:
\[ x = 0 \]