Let e and l are the eccentricity and length of the lactus rectum respectively of the conic described parametrically by `x=t^(2)-t+1, y=t^(2)-t+1`, then the value of `(e )/(l^(2))` is equal to

Answer

To solve the problem, we need to analyze the parametric equations given and identify the conic section they represent. The equations are:

\[ x = t^2 - t + 1 \]
\[ y = t^2 - t + 1 \]

### Step 1: Combine the equations
First, we can combine the two equations to find a relationship between \(x\) and \(y\).

Adding the two equations:
\[ x + y = (t^2 - t + 1) + (t^2 - t + 1) = 2t^2 - 2t + 2 \]
This simplifies to:
\[ x + y = 2(t^2 - t + 1) = 2t^2 - 2t + 2 \]

### Step 2: Subtract the equations
Now, let's subtract the second equation from the first:
\[ x - y = (t^2 - t + 1) - (t^2 - t + 1) = 0 \]
This means:
\[ x - y = 0 \implies x = y \]

### Step 3: Express \(t\) in terms of \(x\) and \(y\)
From the equation \(x + y = 2(t^2 - t + 1)\), we can express \(t\) in terms of \(x\) and \(y\):
\[ x + y = 2t^2 - 2t + 2 \implies t^2 - t + 1 = \frac{x + y}{2} \]

### Step 4: Substitute \(t\) back
Next, we can express \(t\) from \(x - y\):
\[ t = \frac{x - y}{2} \]

### Step 5: Identify the conic section
Now we can substitute \(t\) back into the equations to find the relationship between \(x\) and \(y\). However, since we already know \(x = y\), we can conclude that the conic section is a parabola.

### Step 6: Find the eccentricity and length of the latus rectum
For a parabola, the eccentricity \(e = 1\) and the length of the latus rectum \(l = 4a\).

To find \(a\), we need to identify the vertex of the parabola. The vertex occurs at the minimum point of the quadratic \(t^2 - t + 1\). Completing the square or using the vertex formula gives us \(a = 1\).

Thus, the length of the latus rectum is:
\[ l = 4a = 4 \cdot 1 = 4 \]

### Step 7: Calculate \(\frac{e}{l^2}\)
Now we can calculate the required value:
\[ \frac{e}{l^2} = \frac{1}{4^2} = \frac{1}{16} \]

### Final Answer
The value of \(\frac{e}{l^2}\) is:
\[ \frac{1}{16} \]