Normals at `(x_(1),y_(1)),(x_(2),y_(2))and(x_(3),y_(3))` to the parabola `y^(2)=4x` are concurrent at point P. If `y_(1)y_(2)+y_(2)y_(3)+y_(3)y_(1)=x_(1)x_(2)x_(3)`, then locus of point P is part of parabola, length of whose latus rectum is __________.

Explanation

To solve the problem step by step, we will analyze the conditions given and derive the required locus and its properties.

### Step 1: Understand the Parabola
The equation of the parabola given is \( y^2 = 4x \). This is a standard form of a parabola that opens to the right.

### Step 2: General Equation of the Normal
The normal to the parabola at a point \((x_0, y_0)\) can be expressed as:
\[ y = mx - 2m - m^2 \]
where \(m\) is the slope of the tangent at that point.

### Step 3: Coordinates of Points on the Parabola
Let the points on the parabola be:
- \( (x_1, y_1) = (m_1^2, -2m_1) \)
- \( (x_2, y_2) = (m_2^2, -2m_2) \)
- \( (x_3, y_3) = (m_3^2, -2m_3) \)

### Step 4: Condition of Concurrency
The normals at these points are concurrent at point \(P(h, k)\). The condition given is:
\[ y_1y_2 + y_2y_3 + y_3y_1 = x_1x_2x_3 \]
Substituting the coordinates:
\[ (-2m_1)(-2m_2) + (-2m_2)(-2m_3) + (-2m_3)(-2m_1) = (m_1^2)(m_2^2)(m_3^2) \]
This simplifies to:
\[ 4(m_1m_2 + m_2m_3 + m_3m_1) = m_1^2 m_2^2 m_3^2 \]

### Step 5: Relate the Slopes
Let \(m_1, m_2, m_3\) be the slopes of the normals. From the properties of cubic equations, we know:
- \(m_1m_2 + m_2m_3 + m_3m_1 = -\frac{b}{a}\)
- \(m_1m_2m_3 = -\frac{c}{a}\)

### Step 6: Substitute into the Condition
From the cubic equation formed, we can express:
\[ 4(-\frac{b}{a}) = -\frac{c}{a} \]
This leads us to a relationship between \(h\) and \(k\).

### Step 7: Find the Locus of Point P
After substituting and simplifying, we find:
\[ k^2 = 8 - 4h \]
Rearranging gives us the locus:
\[ k^2 = -4(h - 2) \]
This is a parabola that opens to the left.

### Step 8: Length of the Latus Rectum
The standard form of the parabola is \(y^2 = -4a(x - h)\). Here, \(a = 1\) (since \(4a = 4\)). The length of the latus rectum is given by \(4a\).

Thus, the length of the latus rectum is:
\[ \text{Length of the latus rectum} = 4 \times 1 = 4 \]

### Final Answer
The length of the latus rectum is **4 units**.