The equation of the ellipse with its axes as the coordinate axes and whose latus rectum is 10 and distance between the foci = minor axis is

Options

A. `x^(2)+2y^(2)=16`
B. `x^(2)+2y^(2)=32`
C. `x^(2)+2y^(2)=64`
D. `x^(2)+2y^(2)=100`

Answer

To solve the problem step by step, we need to derive the equation of the ellipse based on the given conditions: the latus rectum is 10 and the distance between the foci equals the minor axis.

### Step 1: Understand the standard form of the ellipse
The standard equation of an ellipse centered at the origin with its axes along the coordinate axes is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \(2a\) is the length of the major axis and \(2b\) is the length of the minor axis.

### Step 2: Use the information about the latus rectum
The latus rectum \(L\) of an ellipse is given by the formula:
\[ L = \frac{2b^2}{a} \]
According to the problem, the latus rectum is 10:
\[ \frac{2b^2}{a} = 10 \]
From this, we can express \(b^2\) in terms of \(a\):
\[ 2b^2 = 10a \implies b^2 = 5a \quad \text{(Equation 1)} \]

### Step 3: Use the information about the distance between the foci
The distance between the foci \(d\) of the ellipse is given by:
\[ d = 2ae \]
where \(e\) is the eccentricity of the ellipse, defined as:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
According to the problem, the distance between the foci is equal to the minor axis \(2b\):
\[ 2ae = 2b \implies ae = b \quad \text{(Equation 2)} \]

### Step 4: Substitute \(e\) in terms of \(a\) and \(b\)
From Equation 2, we can express \(e\):
\[ e = \frac{b}{a} \]
Now substituting this into the eccentricity formula:
\[ \left(\frac{b}{a}\right)^2 = 1 - \frac{b^2}{a^2} \]
This simplifies to:
\[ \frac{b^2}{a^2} = 1 - \frac{b^2}{a^2} \implies 2\frac{b^2}{a^2} = 1 \implies b^2 = \frac{a^2}{2} \quad \text{(Equation 3)} \]

### Step 5: Substitute Equation 1 into Equation 3
From Equation 1, we have \(b^2 = 5a\). Now we can set this equal to Equation 3:
\[ 5a = \frac{a^2}{2} \]
Multiplying both sides by 2 to eliminate the fraction:
\[ 10a = a^2 \]
Rearranging gives:
\[ a^2 - 10a = 0 \]
Factoring out \(a\):
\[ a(a - 10) = 0 \]
Thus, \(a = 0\) (not possible) or \(a = 10\).

### Step 6: Find \(b^2\)
Now substituting \(a = 10\) back into Equation 1 to find \(b^2\):
\[ b^2 = 5a = 5 \times 10 = 50 \]

### Step 7: Write the equation of the ellipse
Now we can substitute \(a^2\) and \(b^2\) into the standard form of the ellipse:
\[ \frac{x^2}{10^2} + \frac{y^2}{50} = 1 \implies \frac{x^2}{100} + \frac{y^2}{50} = 1 \]

### Step 8: Multiply through by 100 to simplify
\[ x^2 + 2y^2 = 100 \]

### Final Answer
The equation of the ellipse is:
\[ x^2 + 2y^2 = 100 \]