The length of the major axis and minor axis of `9x^(2) + y^(2) = 36` respectively are

Options

A. 4 units and 12 units
B. 12 units and 4 units
C. 2 units and 6 units
D. 6 units and 2 units

Answer

To find the lengths of the major and minor axes of the ellipse given by the equation \(9x^2 + y^2 = 36\), we can follow these steps:

### Step 1: Rewrite the equation in standard form
The given equation is:
\[ 9x^2 + y^2 = 36 \]
To convert this into standard form, we divide every term by 36:
\[ \frac{9x^2}{36} + \frac{y^2}{36} = 1 \]
This simplifies to:
\[ \frac{x^2}{4} + \frac{y^2}{36} = 1 \]

### Step 2: Identify the values of \(a\) and \(b\)
The standard form of the ellipse is:
\[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \]
From our equation, we can identify:
- \(b^2 = 4\) (thus, \(b = 2\))
- \(a^2 = 36\) (thus, \(a = 6\))

### Step 3: Calculate the lengths of the axes
The lengths of the axes of an ellipse are given by:
- Length of the major axis = \(2a\)
- Length of the minor axis = \(2b\)

Calculating these:
- Length of the major axis:
\[ 2a = 2 \times 6 = 12 \]
- Length of the minor axis:
\[ 2b = 2 \times 2 = 4 \]

### Conclusion
The lengths of the major and minor axes are:
- Major axis = 12 units
- Minor axis = 4 units

Thus, the lengths of the major axis and minor axis respectively are \(12\) and \(4\).

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