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Question

If the shortest distance(in units) between the parabolas y2=4x and y2=2x6 is d units, then the value of d2 is

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Solution

Shortest distance occurs along common normal
Equation of normal at y2=4x having slope m is
y=mx2mm3 and point of contact is (m2,2m)
For y=mx2mm3=m(x3)+mm3 to be notrmal to y2=2(x3),
mm3=2(12)m12m3
2mm32=0
m=0,±2
Point of contact for two parabolas are (m2,2m) and (3+12m2,m)
For m=0
Point of contact are (0,0) and (3,0), and shortest distance will be 3
For m=2
Point of contact are (4,4) and (5,2), and shortest distance will be 5
For m=2
Point of contact are (4,4) and (5,2), and shortest distance will be 5
shortest distance (d)=5units
d2=5

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