Irrelative

We are given the parabola \( y^2 = \frac{25}{7}x \), and the general equation of a system of parallel chords is \( 4x - y + k = 0 \). We need to find the equation of the corresponding diameter to this system of parallel chords.

### Step 1: Equation of the parabola
The given parabola is \( y^2 = \frac{25}{7}x \), which is in the standard form \( y^2 = 4ax \), where \( a = \frac{25}{28} \).

### Step 2: Equation of the system of parallel chords
The equation of the system of parallel chords is \( 4x - y + k = 0 \), or equivalently, \( y = 4x + k \). This represents a family of lines that are parallel to each other.

### Step 3: Condition for the chords to intersect the parabola
For the line \( y = 4x + k \) to intersect the parabola \( y^2 = \frac{25}{7}x \), we substitute \( y = 4x + k \) into the equation of the parabola:

\[
(4x + k)^2 = \frac{25}{7}x
\]

Expanding the left-hand side:

\[
16x^2 + 8kx + k^2 = \frac{25}{7}x
\]

Multiply the entire equation by 7 to eliminate the fraction:

\[
112x^2 + 56kx + 7k^2 = 25x
\]

Rearrange the equation:

\[
112x^2 + (56k - 25)x + 7k^2 = 0
\]

This is a quadratic equation in \( x \), and for the line to intersect the parabola at two distinct points (i.e., a valid chord), the discriminant of this quadratic equation must be non-negative.

### Step 4: The equation of the diameter
The diameter corresponding to the system of parallel chords can be found by using the property that the diameter is the line passing through the midpoint of the chord. The general formula for the equation of the diameter corresponding to the system of parallel chords \( y = mx + c \) for the parabola \( y^2 = 4ax \) is:

\[
y = mx - \frac{a}{m}
\]

Here, \( m = 4 \) and \( a = \frac{25}{28} \), so the equation of the diameter is:

\[
y = 4x - \frac{\frac{25}{28}}{4} = 4x - \frac{25}{112}
\]

Thus, the equation of the corresponding diameter is:

\[
y = 4x - \frac{25}{112}
\]