The locus of the circumcenter of a variable triangle having sides the y-axis, y=2, and lx+my=1, where (1,m) lies on the parabola `y^(2)=4x`, is a curve C.
The length of the smallest chord of this C is

Options

A. `1//4`
B. `1//12`
C. `1//8`
D. `1//16`

Explanation

To solve the problem, we need to find the locus of the circumcenter of a variable triangle formed by the y-axis, the line \(y = 2\), and the line \(Lx + My = 1\), where the point \((1, m)\) lies on the parabola \(y^2 = 4x\).

### Step 1: Identify the vertices of the triangle
The vertices of the triangle can be identified as follows:
- Point A (on the y-axis at \(y = 2\)): \(A(0, 2)\)
- Point C (intersection of \(Lx + My = 1\) with the y-axis): Set \(x = 0\) in \(Lx + My = 1\) to find \(C(0, \frac{1}{M})\).
- Point B (intersection of \(y = 2\) and \(Lx + My = 1\)): Substitute \(y = 2\) into \(Lx + My = 1\):
\[ Lx + 2M = 1 \implies x = \frac{1 - 2M}{L} \]
So, \(B\left(\frac{1 - 2M}{L}, 2\right)\).

### Step 2: Find the circumcenter
For a right triangle, the circumcenter lies at the midpoint of the hypotenuse. The hypotenuse is the segment \(BC\).

- The coordinates of point B are \(B\left(\frac{1 - 2M}{L}, 2\right)\) and point C is \(C(0, \frac{1}{M})\).
- The midpoint \(P\) of \(BC\) is given by:
\[ H = \frac{\frac{1 - 2M}{L} + 0}{2} = \frac{1 - 2M}{2L}, \quad K = \frac{2 + \frac{1}{M}}{2} = \frac{2M + 1}{2M} \]

### Step 3: Express \(M\) and \(L\) in terms of \(H\) and \(K\)
From the midpoint coordinates, we can express \(M\) and \(L\):
1. From \(K\):
\[ 2KM = 2M + 1 \implies M = \frac{1 - 2K}{2K} \]
2. From \(H\):
\[ 2LH = 1 - 2M \implies L = \frac{1 - 2M}{2H} \]

### Step 4: Substitute into the parabola equation
Given that \((1, m)\) lies on the parabola \(y^2 = 4x\), we have:
\[ m^2 = 4 \cdot 1 \implies m^2 = 4 \]
Thus, \(m = 2\) or \(m = -2\).

### Step 5: Substitute \(M\) and \(L\) into the parabola equation
Substituting \(M\) into the parabola equation:
\[ M^2 = 4L \implies \left(\frac{1 - 2K}{2K}\right)^2 = 4\left(\frac{1 - 2M}{2H}\right) \]
This will yield a relationship between \(H\) and \(K\).

### Step 6: Find the locus equation
After simplification, we can derive the locus equation:
\[ x = 8(y - \frac{3}{2})^2 - \frac{1}{4} \]
This is a parabola in the form \(y^2 = 4ax\).

### Step 7: Determine the length of the smallest chord
The length of the smallest chord of the parabola corresponds to the length of the latus rectum, which is given by:
\[ \text{Length of latus rectum} = 4a \]
From the derived equation, we find \(a = \frac{1}{32}\), thus:
\[ \text{Length} = 4 \times \frac{1}{32} = \frac{1}{8} \]

### Final Answer
The length of the smallest chord of the curve \(C\) is \(\frac{1}{8}\).