Find the equation of the ellipse whose auxilliary circle is the director circle of the ellipse `x^2/36+y^2/13=1` and the length of a latus rectum is 2 units.

Answer

To find the equation of the ellipse whose auxiliary circle is the director circle of the ellipse \( \frac{x^2}{36} + \frac{y^2}{13} = 1 \) and the length of the latus rectum is 2 units, we can follow these steps:

### Step 1: Identify the parameters of the given ellipse
The given ellipse is \( \frac{x^2}{36} + \frac{y^2}{13} = 1 \). From this, we can identify:
- \( a^2 = 36 \) (so \( a = 6 \))
- \( b^2 = 13 \) (so \( b = \sqrt{13} \))

### Step 2: Find the equation of the director circle
The equation of the director circle for an ellipse is given by:
\[ x^2 + y^2 = a^2 + b^2 \]
Substituting the values of \( a^2 \) and \( b^2 \):
\[ x^2 + y^2 = 36 + 13 = 49 \]
Thus, the equation of the director circle is:
\[ x^2 + y^2 = 49 \]

### Step 3: Understand the auxiliary circle of the new ellipse
Let’s denote the new ellipse we need to find as \( \frac{x^2}{\alpha^2} + \frac{y^2}{\beta^2} = 1 \). The auxiliary circle of this ellipse has the equation:
\[ x^2 + y^2 = \alpha^2 \]

### Step 4: Relate the auxiliary circle to the director circle
Since the auxiliary circle of the new ellipse is equal to the director circle of the given ellipse, we have:
\[ \alpha^2 = 49 \]
This gives us:
\[ \alpha = 7 \]

### Step 5: Use the length of the latus rectum to find \( \beta \)
The length of the latus rectum \( L \) for an ellipse is given by:
\[ L = \frac{2b^2}{a} \]
We know from the problem that \( L = 2 \) units. Substituting the values we have:
\[ 2 = \frac{2b^2}{\alpha} \]
Substituting \( \alpha = 7 \):
\[ 2 = \frac{2b^2}{7} \]
Multiplying both sides by 7:
\[ 14 = 2b^2 \]
Dividing by 2:
\[ b^2 = 7 \]

### Step 6: Write the equation of the ellipse
Now that we have \( \alpha^2 = 49 \) and \( \beta^2 = 7 \), the equation of the ellipse is:
\[ \frac{x^2}{49} + \frac{y^2}{7} = 1 \]

### Final Answer:
The equation of the ellipse is:
\[ \frac{x^2}{49} + \frac{y^2}{7} = 1 \]

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