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  • The shortest distance between the parabola  and  <img alt="

The shortest distance between the parabola \mathrm{y^2=4 x} and 

\mathrm{y^2=2 x-6} is 

 

Option: 1

2

 


Option: 2

\sqrt{5}

 


Option: 3

3


Option: 4

none of these 


Answers (1)

best_answer

Shortest distance between two curves occured along the common normal.

Normal to \mathrm{y^2=4 x \text { at }\left(m^2, 2 m\right) \text { is } y+m x-2 m-m^3=0}

Normal to \mathrm{y^2=2(x-3) \text { at }\left(\frac{m^2}{2}+3, m\right)} is 

                 \mathrm{y+m(x-3)-m-\frac{m^3}{2}=0}

Both are same if \mathrm{-2 m-m^3=-4 m-\frac{1}{2} m^3}

\Rightarrow                                          \mathrm{m=0, \pm 2}

So, points will be (4,4) and (5,2) or (4,-4) and (5,-2)

 Hence, shortest distance will be \mathrm{\sqrt{(1+4)}=\sqrt{5}}

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