Maths-
General
Easy

Question

If the points left parenthesis a u to the power of 2 end exponent comma   2 a u right parenthesis and left parenthesis a v to the power of 2 end exponent comma   2 a v right parenthesis are the extremities of a focal chord of the parabola y to the power of 2 end exponent equals 4 a x, then

  1. u v minus 1 equals 0    
  2. u v plus 1 equals 0    
  3. u plus v equals 0    
  4. u minus v equals 0    

The correct answer is: u v plus 1 equals 0


    Equation of focal chord for the parabola y to the power of 2 end exponent equals 4 a x comma passes through the point left parenthesis a u to the power of 2 end exponent comma 2 a u right parenthesis and left parenthesis a v to the power of 2 end exponent comma 2 a v right parenthesis
    Þ y minus 2 a u equals fraction numerator 2 a v minus 2 a u over denominator a v to the power of 2 end exponent minus a u to the power of 2 end exponent end fraction left parenthesis x minus a u to the power of 2 end exponent right parenthesis
    Þ y minus 2 a u equals fraction numerator 2 a left parenthesis v minus u right parenthesis over denominator a left parenthesis v minus u right parenthesis left parenthesis v plus u right parenthesis end fraction left parenthesis x minus a u to the power of 2 end exponent right parenthesis
    Þ y minus 2 a u equals fraction numerator 2 over denominator left parenthesis v plus u right parenthesis end fraction left parenthesis x minus a u to the power of 2 end exponent right parenthesis
    If this is focal chord, so it would passes through focus (a,0).
    Þ0 minus 2 a u equals fraction numerator 2 over denominator v plus u end fraction left parenthesis a minus a u to the power of 2 end exponent right parenthesisÞnegative u v minus u to the power of 2 end exponent equals 1 minus u to the power of 2 end exponent, \uv+1=0.

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