Sponsor Area

Application Of Derivatives

Question
CBSEENMA12035235

Show that the curve xy = a2 and x2 + y2 = 2a2 touch each other.

Solution

The given curves are
                         xy space equals straight a squared                                          ...(1)
              x2 + y2 = 2a2                                             ...(2)
Now left parenthesis straight x plus straight y right parenthesis squared space equals space straight x squared plus straight y squared plus 2 xy space equals space 2 straight a squared plus 2 straight a squared               open square brackets because space of space left parenthesis 1 right parenthesis comma space space left parenthesis 2 right parenthesis close square brackets
therefore space space space space space space left parenthesis straight x plus straight y right parenthesis squared space equals space 4 straight a squared                                                    
rightwards double arrow space space space space space straight x plus straight y equals 2 straight a comma space space minus 2 straight a                                         ...(3)
Also  left parenthesis straight x minus straight y right parenthesis squared space equals space straight x squared plus straight y squared minus 2 xy
                        equals 2 straight a squared minus 2 straight a squared space space space space space space space space space space space space space space space space                           open square brackets because space space space of space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis close square brackets
                         =0
therefore space space space space space space straight x minus straight y space equals space 0                                                      ...(4)
Adding (3) and (4),  we get,
                     2 straight x space equals 2 straight a comma space space space space space space minus 2 straight a space space space space space rightwards double arrow space space space space straight x space equals space straight a comma space space space minus straight a
therefore space space space space from space left parenthesis 4 right parenthesis comma space space space space straight y space equals space straight a comma space space space space minus straight a
therefore space space space curves space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis space intersect space at space left parenthesis straight a comma space straight a right parenthesis space and space left parenthesis negative straight a comma space minus straight a right parenthesis
From space left parenthesis 1 right parenthesis comma space straight y space equals space straight a squared over straight x
therefore space space space space dy over dx space equals space minus straight a squared over straight x squared
From space left parenthesis 2 right parenthesis comma space space space 2 straight x plus 2 straight y dy over dx space equals space 0
rightwards double arrow space space space space space space dy over dx space equals space minus straight x over straight y
Let space space straight m subscript 1 comma space space straight m subscript 2 space be space slopes space of space curves space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis
At space left parenthesis straight a comma space straight a right parenthesis
space space space space straight m subscript 1 space equals space minus straight a squared over straight a squared equals negative 1 comma space space space space space straight m subscript 2 space equals space minus straight a over straight a space equals space minus 1
therefore space space at space left parenthesis straight a comma space straight a right parenthesis space curves space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis space touch space
At space left parenthesis negative straight a comma space minus straight a right parenthesis
space space space space straight m subscript 1 space equals space minus straight a squared over straight a squared space equals space minus 1 comma space space space straight m subscript 2 space equals space minus fraction numerator negative straight a over denominator negative straight a end fraction equals negative 1
therefore space space space at space left parenthesis negative straight a comma space space space minus straight a right parenthesis comma space curves space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis space touch.