Focus (3, 0), e =`3/5`, d irectrix 3x-25= 0, equation of the ellipse is

Options

A. `(x^(2))/(16)+(y^(2))/(25)=1`
B. `(x^(2))/(25)+(y^(2))/(16)=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 given the focus, eccentricity, and directrix, we can follow these steps:

### Step 1: Identify the given data
- Focus: \( (3, 0) \)
- Eccentricity: \( e = \frac{3}{5} \)
- Directrix: \( 3x - 25 = 0 \) or \( x = \frac{25}{3} \)

### Step 2: Determine the value of \( a \)
The formula relating the distance from the focus to the directrix and the eccentricity is given by:
\[ \frac{a}{e} = \text{Distance from focus to directrix} \]
The distance from the focus \( (3, 0) \) to the directrix \( x = \frac{25}{3} \) is:
\[ \text{Distance} = \left| 3 - \frac{25}{3} \right| = \left| \frac{9}{3} - \frac{25}{3} \right| = \left| \frac{-16}{3} \right| = \frac{16}{3} \]
Now, substituting the values into the formula:
\[ \frac{a}{\frac{3}{5}} = \frac{16}{3} \]
Cross-multiplying gives:
\[ 5a = \frac{16 \cdot 3}{3} \Rightarrow 5a = 16 \Rightarrow a = \frac{16}{5} \]

### Step 3: Determine the value of \( c \)
Since the focus is at \( (3, 0) \), we have:
\[ c = 3 \]

### Step 4: Calculate \( b \)
Using the relationship \( b^2 = a^2 - c^2 \):
\[ a^2 = \left( \frac{16}{5} \right)^2 = \frac{256}{25} \]
\[ c^2 = 3^2 = 9 = \frac{225}{25} \]
Now substituting these into the equation for \( b^2 \):
\[ b^2 = \frac{256}{25} - \frac{225}{25} = \frac{31}{25} \]

### Step 5: Write the equation of the ellipse
The standard form of the equation of the ellipse with a horizontal major axis is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Substituting \( a^2 \) and \( b^2 \):
\[ \frac{x^2}{\frac{256}{25}} + \frac{y^2}{\frac{31}{25}} = 1 \]
Multiplying through by \( 25 \) to eliminate the denominators:
\[ \frac{25x^2}{256} + \frac{25y^2}{31} = 25 \]

### Final Equation
The final equation of the ellipse is:
\[ \frac{25x^2}{256} + \frac{25y^2}{31} = 25 \]