Find p and q so that y = 2 x is a tangent, line at x = 3, to the curvey = p x 2 + q x + 2

Dear nitin,

• The point of tangency is on the tangent line y = 2x, hence its y coordinate can also be calculated and its coordinates given by
  
  (3 , 2(3)) = (3 , 6)
  
  
• The above point of tangency is also on the curve
  y = p x ² + q x + 2, hence
  
  6 = p(3) ² + q (3) + 2
  
  
• Let us now find y '
  
  y ' = 2 p x + q
  
  
• At x = 3 y ' is equal to the slope of the tangent line y = 2 x. Hence
  
  2 = 2 p (3) + q
  
  
• We now solve the two linear equations
  
  9 p + 3 q = 4 and 6 p + q = 2
  
  
• to find 9 p + 3 q = 4 and 6 p + q = 2
  
  p = 2 / 9 and q = 2 / 3

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