If the distance between foci is 2 and the distance between the directrices is 5, then equation of the ellipse in the standard form is

Options

A. `6x^2+10y^2=5`
B. `6x^2+10y^2=15`
C. `x^2+3y^2=15`
D. `3x^2+3y^2=15`

Explanation

To find the equation of the ellipse in standard form given the distance between the foci and the distance between the directrices, we can follow these steps:

### Step 1: Understand the given information
- The distance between the foci is given as 2.
- The distance between the directrices is given as 5.

### Step 2: Relate the given distances to the ellipse parameters
For an ellipse, the distance between the foci is given by \(2c\) where \(c = ae\) (where \(a\) is the semi-major axis and \(e\) is the eccentricity). Therefore, we have:
\[ 2c = 2 \implies c = 1 \quad \text{(Equation 1)} \]

The distance between the directrices is given by \(2 \cdot \frac{a}{e}\). Therefore, we have:
\[ 2 \cdot \frac{a}{e} = 5 \implies \frac{a}{e} = \frac{5}{2} \quad \text{(Equation 2)} \]

### Step 3: Use the relationships to find \(a\) and \(e\)
From Equation 1, we know:
\[ c = ae = 1 \quad \text{(Equation 3)} \]

Now we can express \(e\) in terms of \(a\) using Equation 2:
\[ e = \frac{a}{\frac{5}{2}} = \frac{2a}{5} \]

### Step 4: Substitute \(e\) in Equation 3
Substituting \(e\) into Equation 3:
\[ 1 = a \cdot \frac{2a}{5} \]
\[ 1 = \frac{2a^2}{5} \]
Multiplying both sides by 5:
\[ 5 = 2a^2 \]
Dividing by 2:
\[ a^2 = \frac{5}{2} \quad \text{(Equation 4)} \]

### Step 5: Find \(b^2\)
Using the relationship \(b^2 = a^2(1 - e^2)\), we first need to find \(e^2\):
\[ e = \frac{2a}{5} \implies e^2 = \left(\frac{2a}{5}\right)^2 = \frac{4a^2}{25} \]
Substituting \(a^2\) from Equation 4:
\[ e^2 = \frac{4 \cdot \frac{5}{2}}{25} = \frac{10}{25} = \frac{2}{5} \]
Now substituting \(e^2\) into the equation for \(b^2\):
\[ b^2 = a^2(1 - e^2) = \frac{5}{2} \left(1 - \frac{2}{5}\right) = \frac{5}{2} \cdot \frac{3}{5} = \frac{3}{2} \quad \text{(Equation 5)} \]

### Step 6: Write the equation of the ellipse
The standard form of the ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Substituting \(a^2\) and \(b^2\) from Equations 4 and 5:
\[ \frac{x^2}{\frac{5}{2}} + \frac{y^2}{\frac{3}{2}} = 1 \]
Multiplying through by 2 to eliminate the fractions:
\[ \frac{2x^2}{5} + \frac{2y^2}{3} = 1 \]
Rearranging gives:
\[ 6x^2 + 10y^2 = 15 \]

### Final Answer
The equation of the ellipse in standard form is:
\[ 6x^2 + 10y^2 = 15 \]