Length of the major axis of the ellipse `9x^(2)+7y^(2)=63,` is

Options

A. 3
B. 9
C. 6
D. `2sqrt(17)`

Explanation

To find the length of the major axis of the ellipse given by the equation \(9x^2 + 7y^2 = 63\), we will follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the equation of the ellipse:
\[ 9x^2 + 7y^2 = 63 \]
To convert this into standard form, we divide the entire equation by 63:
\[ \frac{9x^2}{63} + \frac{7y^2}{63} = 1 \]
This simplifies to:
\[ \frac{x^2}{7} + \frac{y^2}{9} = 1 \]

### Step 2: Identify the values of \(a^2\) and \(b^2\)
In the standard form of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we can identify:
- \(a^2 = 7\)
- \(b^2 = 9\)

### Step 3: Determine \(a\) and \(b\)
Now we find \(a\) and \(b\):
\[ a = \sqrt{7} \quad \text{and} \quad b = \sqrt{9} = 3 \]

### Step 4: Identify the orientation of the ellipse
Since \(b > a\) (i.e., \(3 > \sqrt{7}\)), the major axis is along the y-axis.

### Step 5: Calculate the length of the major axis
The length of the major axis of an ellipse is given by:
\[ \text{Length of major axis} = 2b \]
Substituting the value of \(b\):
\[ \text{Length of major axis} = 2 \times 3 = 6 \]

### Final Answer
Thus, the length of the major axis of the ellipse is \(6\).

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