The co-ordinates of the vertices of the ellipse `(X^(2))/(16) + (y^(2))/(9) =1` are

Options

A. `(pm 4, 0)`
B. `(0, pm 4)`
C. `(pm3, 0)`
D. `(0, pm 3)`

Answer

To find the coordinates of the vertices of the ellipse given by the equation \(\frac{x^2}{16} + \frac{y^2}{9} = 1\), we will follow these steps:

### Step 1: Identify the standard form of the ellipse
The standard form of an ellipse centered at the origin is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \(a^2\) and \(b^2\) are the squares of the semi-major and semi-minor axes, respectively.

### Step 2: Compare the given equation with the standard form
From the given equation:
\[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \]
we can identify:
- \(a^2 = 16\)
- \(b^2 = 9\)

### Step 3: Calculate the values of \(a\) and \(b\)
To find \(a\) and \(b\), we take the square roots:
\[ a = \sqrt{16} = 4 \]
\[ b = \sqrt{9} = 3 \]

### Step 4: Determine the orientation of the ellipse
Since \(a > b\) (4 > 3), the major axis of the ellipse is along the x-axis.

### Step 5: Find the coordinates of the vertices
For an ellipse with the major axis along the x-axis, the coordinates of the vertices are given by:
\[ (\pm a, 0) \]
Substituting the value of \(a\):
\[ (\pm 4, 0) \]
Thus, the coordinates of the vertices are:
\[ (4, 0) \quad \text{and} \quad (-4, 0) \]

### Final Answer:
The coordinates of the vertices of the ellipse are \((4, 0)\) and \((-4, 0)\).

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