Folding a parabola

As shown in the figure, we have the parabola $y = x^2$ . Now, we draw the line passing through $(-\frac{1}{4} , 1)$ (the red dot) and making an angle of $\frac{\pi}{12}$ with the positive $x$ -axis (the black chord). Then we fold the lower part of the parabola such that the folded part becomes tangent to the the black chord at the red dot. Find the length of fold line (the green line segment). This problem requires the use of a numerical method. The image below does not represent the actual values specified above.

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