If one vertex of an ellipse is (0,7) and the corresponding directrix is y = 12 , then the equation of the ellipse is

Options

A. `95x^(2) + 144y^(2) = 4655`
B. `144x^(2)+95y^(2) = 4655`
C. `95x^(2) + 144y^(2) = 13680`
D. none of these

Answer

To find the equation of the ellipse given one vertex at (0, 7) and the corresponding directrix at y = 12, we can follow these steps:

### Step 1: Identify the values of \( a \) and \( e \)
The vertex of the ellipse is given as (0, 7). In the standard form of an ellipse, the vertex is located at (0, a). Thus, we have:
- \( a = 7 \)

The directrix is given by the equation \( y = 12 \). The relationship between the vertex, directrix, and the eccentricity \( e \) is given by:
\[ \frac{a}{e} = 12 \]
Substituting the value of \( a \):
\[ \frac{7}{e} = 12 \]
From this, we can solve for \( e \):
\[ e = \frac{7}{12} \]

### Step 2: Find the value of \( c \)
The eccentricity \( e \) is also defined as:
\[ e = \frac{c}{a} \]
Substituting the known values:
\[ \frac{c}{7} = \frac{7}{12} \]
Cross-multiplying gives:
\[ c = \frac{7 \times 7}{12} = \frac{49}{12} \]

### Step 3: Use the relationship \( a^2 = b^2 + c^2 \)
We know:
- \( a^2 = 7^2 = 49 \)
- \( c^2 = \left(\frac{49}{12}\right)^2 = \frac{2401}{144} \)

Using the relationship:
\[ a^2 = b^2 + c^2 \]
Substituting the known values:
\[ 49 = b^2 + \frac{2401}{144} \]
To solve for \( b^2 \), first convert 49 into a fraction with a denominator of 144:
\[ 49 = \frac{49 \times 144}{144} = \frac{7056}{144} \]
Now, substituting:
\[ \frac{7056}{144} = b^2 + \frac{2401}{144} \]
Subtracting \( \frac{2401}{144} \) from both sides:
\[ b^2 = \frac{7056 - 2401}{144} = \frac{4655}{144} \]

### Step 4: Write the equation of the ellipse
Since the major axis is along the y-axis, the standard form of the ellipse is:
\[ \frac{y^2}{a^2} + \frac{x^2}{b^2} = 1 \]
Substituting \( a^2 \) and \( b^2 \):
\[ \frac{y^2}{49} + \frac{x^2}{\frac{4655}{144}} = 1 \]
To eliminate the fraction in \( b^2 \), multiply through by \( 144 \):
\[ \frac{144y^2}{49} + \frac{144x^2}{\frac{4655}{144}} = 144 \]
This simplifies to:
\[ 144y^2 + \frac{144 \cdot 144 x^2}{4655} = 144 \cdot 49 \]
Thus, the equation of the ellipse can be expressed as:
\[ 95y^2 + 144x^2 = 4655 \]

### Final Equation
The final equation of the ellipse is:
\[ 95y^2 + 144x^2 = 4655 \]