Question

The girder of a railway bridge is a parabola with its vertex at the highest point, 10metre above the ends.
Its span is 100 metres.
i. Find the co-ordinates of the focus of the parabola
ii. Find the equation of bridge
iii. Find the length of latus rectum
iv. Find the height of the bridge at 20 metres from the mid point.
v. Find the equation of circle with centre at focus of the parabola and passes through the vertex of the
parabola.

Answer 1

Given : The girder bridge of a railway bridge is in the parabolic form with span 100 m and the highest point on the arch is 10 m above the bridge

To find : height of the bridge at 20 m to the left or right from the midpoint of the bridge​

Solution:

y = ax² + bx  + c

at x = 0   y = 10

=> c  = 10

y = ax² + bx  + 10

span 100ft

Hence x = 50  & -50   for  y = 0

=> 2500a  + 50b   + 10 = 0

   2500a   - 50b  + 10 = 0

=> 5000a + 20 = 0

=> a  = -20/5000

=>a = -1/250

100b = 0

=> b = 0

y  =  -x²/250   + 10

20 m  left or right from the midpoint of the bridge​

=> x = -20 or 20

=> y =  - (20)²/250  + 10

=> y = -400/250 + 10

=> y =  -1.6 + 10

=> y = 8.4

height of the bridge at 20 m  from the midpoint of the bridge​ = 8.4 m

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