The coordinate of foci of the ellipse `x^2/25+y^2/9=1` is

Options

A. (-4,0) and (4,0)
B. (-3,0) and (3,0)
C. (-9,0) and(9,0)
D. (-5,0) and (5,0)

Answer

To find the coordinates of the foci of the ellipse given by the equation \( \frac{x^2}{25} + \frac{y^2}{9} = 1 \), we can follow these steps:

### Step 1: Identify the values of \( a^2 \) and \( b^2 \)
The standard form of the equation of an ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
From the given equation, we can identify:
- \( a^2 = 25 \) (which implies \( a = \sqrt{25} = 5 \))
- \( b^2 = 9 \) (which implies \( b = \sqrt{9} = 3 \))

### Step 2: Determine the orientation of the ellipse
Since \( a^2 > b^2 \), the major axis of the ellipse is along the x-axis.

### Step 3: Calculate the eccentricity \( e \)
The eccentricity \( e \) of an ellipse is given by the formula:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Substituting the values of \( a^2 \) and \( b^2 \):
\[ e = \sqrt{1 - \frac{9}{25}} = \sqrt{1 - 0.36} = \sqrt{0.64} = 0.8 \]

### Step 4: Calculate \( ae \)
The distance from the center to each focus (denoted as \( ae \)) is calculated as:
\[ ae = a \cdot e = 5 \cdot 0.8 = 4 \]

### Step 5: Write the coordinates of the foci
For an ellipse with a horizontal major axis, the coordinates of the foci are given by:
\[ (\pm ae, 0) \]
Substituting the value of \( ae \):
\[ (\pm 4, 0) \]
Thus, the coordinates of the foci are \( (4, 0) \) and \( (-4, 0) \).

### Final Answer
The coordinates of the foci of the ellipse \( \frac{x^2}{25} + \frac{y^2}{9} = 1 \) are \( (4, 0) \) and \( (-4, 0) \).

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