Q.The tower of a bridge, hung in the form of a parabola have their tops 30 meters above the road way and are 200 meters apart. If the cable is 5 meters above the road way at the centre of the bridge, find the length of the vertical supporting cable from the centre.

Take the origin at the lowest point of the cable (the centre of the bridge), with the $X$-axis horizontal along the roadway and the $Y$-axis vertical. The cable is a parabola $x^2=4by$ opening upward, with $y$ measured from the lowest point.

The towers are $200$ m apart, so each tower is at $x=\pm100$ from the centre. The tops are $30$ m above the road, while the cable's lowest point is $5$ m above the road, so the cable rises $30-5=25$ m from its lowest point to the tower.

So the point $(100,25)$ lies on $x^2=4by$: $100^2=4b(25) \Rightarrow 10000=100b \Rightarrow b=100$.

Equation of the cable: $x^2=400y$. …

Set up a parabola with vertex at the cable's lowest point, use the tower's known coordinates to find the one unknown parameter, then express the h …

Placing the origin at the road level instead of at the cable's lowest point, which makes the …