Step-by-step Solution:

Step 1: Equation of Chord using Midpoint Formula
The equation of a chord of a parabola \( y^2 = 4ax \) with midpoint \( (h, k) \) is given by the midpoint formula: \[ T = 0 \Rightarrow yy_1 = 2(x + x_1) \] Since the chord passes through the vertex (0,0), substituting \( x = 0, y = 0 \) in the chord equation: \[ 0 = 2(h + x_1) \Rightarrow x_1 = -h \] \[ yy_1 = 2(x + x_1) \Rightarrow yk = 2(x - h) \] Replacing \( x_1, y_1 \) with \( (-h, -k) \) (as midpoint formula uses symmetry), \[ k(-k) = 2(h - h) \] \[ -k^2 = -2h \] \[ h = \frac{k^2}{2} \] Step 2: Finding the Locus
Replacing \( h \) with \( x \) and \( k \) with \( y \), we get: \[ x = \frac{y^2}{2} \] \[ y^2 = 2x \] Step 3: Conclusion
Thus, the required locus is: \( {y^2 = 2x} \)