The eccentricity of an ellipse, with centre at the origin is 2/3. If one of directrices is x = 6, then equation of the ellipse is

Options

A. `x^2/9+y^2/4=1`
B. `x^2/9+y^2/5=1`
C. `5x^2+9y^2=80`
D. `3x^2+2y^2=6`

Answer

To find the equation of the ellipse given its eccentricity and one of its directrices, we can follow these steps:

### Step 1: Understand the given information
We know the eccentricity \( e \) of the ellipse is \( \frac{2}{3} \), and one of the directrices is \( x = 6 \).

### Step 2: Use the relationship between the directrix and the semi-major axis
The formula for the directrix of an ellipse is given by:
\[ x = \frac{a}{e} \]
where \( a \) is the semi-major axis and \( e \) is the eccentricity.

Given that one of the directrices is \( x = 6 \), we can set up the equation:
\[ \frac{a}{e} = 6 \]

### Step 3: Substitute the value of eccentricity
Substituting \( e = \frac{2}{3} \) into the equation:
\[ \frac{a}{\frac{2}{3}} = 6 \]
This simplifies to:
\[ a \cdot \frac{3}{2} = 6 \]
Multiplying both sides by \( \frac{2}{3} \):
\[ a = 6 \cdot \frac{2}{3} = 4 \]

### Step 4: Find \( b^2 \) using the eccentricity formula
The eccentricity \( e \) is also related to \( a \) and \( b \) (the semi-minor axis) by the formula:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Substituting \( e = \frac{2}{3} \) and \( a = 4 \):
\[ \frac{2}{3} = \sqrt{1 - \frac{b^2}{4^2}} \]
Squaring both sides:
\[ \left(\frac{2}{3}\right)^2 = 1 - \frac{b^2}{16} \]
\[ \frac{4}{9} = 1 - \frac{b^2}{16} \]
Rearranging gives:
\[ \frac{b^2}{16} = 1 - \frac{4}{9} \]
Finding a common denominator (9):
\[ 1 = \frac{9}{9} \implies 1 - \frac{4}{9} = \frac{5}{9} \]
Thus:
\[ \frac{b^2}{16} = \frac{5}{9} \]
Multiplying both sides by 16:
\[ b^2 = \frac{5 \cdot 16}{9} = \frac{80}{9} \]

### Step 5: Write the equation of the ellipse
The standard form of the ellipse centered at the origin is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Substituting \( a^2 = 4^2 = 16 \) and \( b^2 = \frac{80}{9} \):
\[ \frac{x^2}{16} + \frac{y^2}{\frac{80}{9}} = 1 \]
To eliminate the fraction in the second term, multiply through by 16:
\[ \frac{x^2}{16} + \frac{9y^2}{80} = 1 \]
This can be rearranged to:
\[ 5x^2 + 9y^2 = 80 \]

### Final Answer
The equation of the ellipse is:
\[ 5x^2 + 9y^2 = 80 \]