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Q.
On the parabola $ y={{x}^{2}} $ , the point least distant from the straight line $ y=2x-4 $ is
Jharkhand CECEJharkhand CECE 2014
Solution:
Given, parabola is $ y={{x}^{2}} $ ... (i)
straight line is $ y=2x-4 $ ... (ii) From Eqs. (i) and (ii), we get
$ {{x}^{2}}-2x-4=0 $
Let $ f(x)={{x}^{2}}-2x-4 $
$ \therefore $ $ f'(x)=2x-2 $
For least distance, put $ f'(x)=0 $
$ \Rightarrow $ $ 2x-2=0 $
$ \Rightarrow $ $ x=1 $ From Eq. (i), $ y=1 $
Hence, the point least distant from the line is $ (1,\,\,1). $