If the latus rectum of an ellipse with major axis along y-axis and centre at origin is `(1)/(5)`, distance between foci = length of minor axis, then the equation of the ellipse is

Options

A. `50y^(2) + 25x^(2) =1`
B. `(x^(2))/(50) + (y^(2))/(25) =1`
C. `(x^(2))/(25) + (y^(2))/(50) =1`
D. `50x^(2) + 25y^(2) =1`

Answer

To find the equation of the ellipse given the conditions, we will follow these steps:

### Step 1: Understand the properties of the ellipse
Since the major axis is along the y-axis and the center is at the origin, the standard form of the ellipse is given by:
\[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \]
where \( a \) is the semi-major axis and \( b \) is the semi-minor axis.

### Step 2: Use the length of the latus rectum
The length of the latus rectum \( L \) for an ellipse is given by:
\[ L = \frac{2a^2}{b} \]
According to the problem, the length of the latus rectum is \( \frac{1}{5} \). Therefore, we can set up the equation:
\[ \frac{2a^2}{b} = \frac{1}{5} \]
From this, we can express \( b \) in terms of \( a \):
\[ 2a^2 = \frac{1}{5}b \implies b = 10a^2 \]

### Step 3: Relate the distance between foci to the length of the minor axis
The distance between the foci \( 2c \) is given by:
\[ c = \sqrt{a^2 - b^2} \]
The problem states that the distance between the foci is equal to the length of the minor axis, which is \( 2b \). Thus, we have:
\[ 2c = 2b \implies c = b \]
Substituting for \( c \):
\[ \sqrt{a^2 - b^2} = b \]
Squaring both sides gives:
\[ a^2 - b^2 = b^2 \implies a^2 = 2b^2 \]

### Step 4: Substitute for \( b \)
From step 2, we have \( b = 10a^2 \). Substituting this into the equation \( a^2 = 2b^2 \):
\[ a^2 = 2(10a^2)^2 \]
This simplifies to:
\[ a^2 = 2 \cdot 100a^4 \implies a^2 = 200a^4 \]
Rearranging gives:
\[ 200a^4 - a^2 = 0 \implies a^2(200a^2 - 1) = 0 \]
Thus, \( a^2 = 0 \) or \( 200a^2 = 1 \). Since \( a^2 \) cannot be zero, we have:
\[ a^2 = \frac{1}{200} \]

### Step 5: Find \( b^2 \)
Using \( b = 10a^2 \):
\[ b^2 = 10^2 \cdot a^4 = 100 \cdot \left(\frac{1}{200}\right)^2 = 100 \cdot \frac{1}{40000} = \frac{1}{400} \]

### Step 6: Write the equation of the ellipse
Now substituting \( a^2 \) and \( b^2 \) into the standard form of the ellipse:
\[ \frac{x^2}{\frac{1}{400}} + \frac{y^2}{\frac{1}{200}} = 1 \]
Multiplying through by 400 gives:
\[ 400x^2 + 2y^2 = 1 \]
or
\[ 200x^2 + y^2 = \frac{1}{2} \]

### Final Equation
Thus, the equation of the ellipse is:
\[ 200x^2 + y^2 = 1 \]