The equation of the ellipse whose length of the major axis is 10 units and co-ordinates of the foci are `(0, pm 4)` is

Options

A. `(x^(2))/(5) + (y^(2))/(3) =1`
B. `(x^(2))/(3) + (y^(2))/(5) =1`
C. `(x^(2))/(25) + (y^(2))/(9) =1`
D. `(x^(2))/(9) + (y^(2))/(25) =1`

Answer

To find the equation of the ellipse with the given parameters, we can follow these steps:

### Step 1: Identify the lengths of the axes
The length of the major axis is given as 10 units. Therefore, the semi-major axis \( b \) can be calculated as:
\[ b = \frac{\text{Length of Major Axis}}{2} = \frac{10}{2} = 5 \]

### Step 2: Identify the coordinates of the foci
The coordinates of the foci are given as \( (0, \pm 4) \). This indicates that the foci are located on the y-axis, which means the major axis is along the y-axis.

### Step 3: Determine the relationship between a, b, and c
In an ellipse, the relationship between the semi-major axis \( b \), semi-minor axis \( a \), and the distance to the foci \( c \) is given by:
\[ c^2 = b^2 - a^2 \]
From the foci coordinates, we have \( c = 4 \).

### Step 4: Substitute the known values into the equation
We know \( b = 5 \) and \( c = 4 \). We can substitute these values into the equation:
\[ 4^2 = 5^2 - a^2 \]
This simplifies to:
\[ 16 = 25 - a^2 \]

### Step 5: Solve for \( a^2 \)
Rearranging the equation gives:
\[ a^2 = 25 - 16 = 9 \]
Thus, we find:
\[ a = \sqrt{9} = 3 \]

### Step 6: Write the equation of the ellipse
Since the major axis is along the y-axis, the standard form of the equation of the ellipse is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Substituting the values of \( a \) and \( b \):
\[ \frac{x^2}{3^2} + \frac{y^2}{5^2} = 1 \]
This simplifies to:
\[ \frac{x^2}{9} + \frac{y^2}{25} = 1 \]

### Final Answer
The equation of the ellipse is:
\[ \frac{x^2}{9} + \frac{y^2}{25} = 1 \]

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