Find The Length

This is a nice question which is based on the property that the foot of perpendicular from focus of parabola to any tangent lies on the tangent at the vertex .

The diagram is not according to scale.

I am not posting the complete solution but i can provide the steps.

1) Find the point of intersection of $3x+4y=7$ and $y=x$ , also of $2x+3y=10$ and $y=x$

2) Point of intersection are $A$ and $B$

3) find the equation of $AS$ and $BS$ which are perpendicular to respective tangents of parabola.

4) The point of intersection of $AS$ and $BS$ gives the coordinates of focus.

5) length of perpendicular from focus to the tangent at the vertex gives $a$ .

6) Just calculate $4a$