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KCET2010MathematicsParabola

The condition for the line y=m x+c to be a normal to the parabola y ²=4 ax is

Options

  1. Ac =-2 am -a ~m ³
  2. Bc=- a m
  3. Cc = a m
  4. Dc=2 a m+a m³

Correct answer

A. c =-2 am -a ~m ³

Step-by-step solution

Given that, equation of parabola y²=4 a x , let the parametric coordinate is ( am ², 2 am ) . array ll & 2 y d y d x =4 a & d y d x = 2 a y array Slope of normal = ( -y 2 a ) At ( am ², 2 am )= -2 am 2 a =- m Now, the equation of normal to the parabola is gathered ( y -2 am )=(- m ) ( x - am ² ) y -2 am =- mx +a ~m ³ mx + y - (2 am +a ~m ³ )=0 ...(i) gathered Also, given the line y=m x+c or m x-y+c=0 ...(ii) is normal to parabola, then On comparing c =-2 am - am ³

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