An ellipse has it centre at (1,-1) and semimajor axis=8 and which passes through the point (1,3) if l be length of its latus rectum then `(l)/(4)` is ______

Answer

To solve the problem step by step, we will derive the equation of the ellipse, find the value of \( b^2 \), calculate the length of the latus rectum, and finally compute \( \frac{L}{4} \).

### Step 1: Write the standard form of the ellipse
The standard form of an ellipse centered at \((h, k)\) with a semi-major axis \(a\) and semi-minor axis \(b\) is given by:
\[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \]
Given that the center is \((1, -1)\) and the semi-major axis \(a = 8\), we can substitute these values into the equation:
\[ \frac{(x - 1)^2}{8^2} + \frac{(y + 1)^2}{b^2} = 1 \]
This simplifies to:
\[ \frac{(x - 1)^2}{64} + \frac{(y + 1)^2}{b^2} = 1 \]

### Step 2: Substitute the point (1, 3) into the ellipse equation
Since the ellipse passes through the point \((1, 3)\), we substitute \(x = 1\) and \(y = 3\) into the equation:
\[ \frac{(1 - 1)^2}{64} + \frac{(3 + 1)^2}{b^2} = 1 \]
This simplifies to:
\[ 0 + \frac{4^2}{b^2} = 1 \]
\[ \frac{16}{b^2} = 1 \]
Multiplying both sides by \(b^2\) gives:
\[ 16 = b^2 \]
Thus, we find:
\[ b = \sqrt{16} = 4 \]

### Step 3: Calculate the length of the latus rectum
The length of the latus rectum \(L\) of an ellipse is given by the formula:
\[ L = \frac{2b^2}{a} \]
Substituting the values of \(b^2\) and \(a\):
\[ L = \frac{2 \times 16}{8} = \frac{32}{8} = 4 \]

### Step 4: Find \(\frac{L}{4}\)
Now, we compute \(\frac{L}{4}\):
\[ \frac{L}{4} = \frac{4}{4} = 1 \]

### Final Answer
Thus, the value of \(\frac{L}{4}\) is:
\[ \boxed{1} \]