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AP EAMCET201724 Apr 2017Evening ShiftMathematicsApplication of DerivativesActual

The shortest distance between the line y-x=1 and the curve x=y^2 is

Options

  1. A2 3 8
  2. B3 2 5
  3. C3 4
  4. D3 2 8

Correct answer

D. 3 2 8

Step-by-step solution

The shortest distance between the line and curve occurs when the tangent to the curve is parallel to the given line. Rewriting the line y-x=1 in standard form gives x-y+1=0 , which has slope m_L = 1 . For the curve x=y^2 , differentiating yields dx dy = 2y , so the tangent slope is dy dx = 1 2y . Setting this equal to the line's slope gives 1 2y = 1 , so y = 1 2 . The corresponding x-coordinate is x = y^2 = 1 4 . The distance from point ( 1 4 , 1 2 ) to the line x-y+1=0 is: D = |1 1 4 -1 1 2 +1 | 1^2 + (-1)^2 = 3 4

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