Application of Derivatives

Find the equations of the tangent and normal to the curve - Wired Faculty

Question

Find the equations of the tangent and normal to the curve $straight x squared over straight a squared minus straight y squared over straight b squared equals 1 space at space the space point space open parentheses square root of 2 straight a comma space straight b close parentheses.$

Solution

Let $straight x squared over straight a squared minus straight y squared over straight b squared equals 1$ be the equation of the curve.
Rewriting the above equation as,
$straight y squared over straight b squared equals straight x squared over straight a squared minus 1 space rightwards double arrow space space space straight y squared space equals space straight b squared over straight a squared straight x squared minus straight b squared$
Differentiating the above function with respect to x, we get,
$2 straight y dy over dx equals straight b squared over straight a squared 2 straight x rightwards double arrow space space dy over dx equals straight b squared over straight a squared straight x over straight y rightwards double arrow space space open square brackets dy over dx close square brackets subscript left parenthesis square root of 2 straight a comma space straight b right parenthesis end subscript space equals space straight b squared over straight a squared fraction numerator square root of 2 straight a over denominator straight b end fraction equals fraction numerator square root of 2 straight b over denominator straight a end fraction$
Slope of the tangent is $straight m equals fraction numerator square root of 2 straight b over denominator straight a end fraction$
Equation of the tangent is
$left parenthesis straight y minus straight y subscript 1 right parenthesis space equals space straight m left parenthesis straight x minus straight x subscript 1 right parenthesis rightwards double arrow space space space left parenthesis straight y minus straight b right parenthesis equals fraction numerator square root of 2 straight b over denominator straight a end fraction open parentheses straight x minus square root of 2 straight a close parentheses rightwards double arrow space space straight a left parenthesis straight y minus straight b right parenthesis equals square root of 2 straight b left parenthesis straight x minus square root of 2 straight a right parenthesis rightwards double arrow space square root of 2 bx minus ay plus ab minus 2 ab equals 0 rightwards double arrow square root of 2 bx minus ay minus ab space equals space 0$
Slope of the normal is $negative fraction numerator 1 over denominator begin display style fraction numerator square root of 2 straight b over denominator straight a end fraction end style end fraction$
Equation of the normal is
$open parentheses straight y minus straight y subscript 1 close parentheses space equals space straight m left parenthesis straight x minus straight x subscript 1 right parenthesis rightwards double arrow space space left parenthesis straight y minus straight b right parenthesis space equals space fraction numerator negative straight a over denominator square root of 2 straight b end fraction left parenthesis straight x minus square root of 2 straight a right parenthesis rightwards double arrow square root of 2 straight b left parenthesis straight y minus straight b right parenthesis equals negative straight a left parenthesis straight x minus square root of 2 straight a right parenthesis rightwards double arrow ax plus square root of 2 by minus square root of 2 straight b squared plus square root of 2 straight a squared equals 0 rightwards double arrow ax plus square root of 2 by plus square root of 2 left parenthesis straight a squared minus straight b squared right parenthesis space equals space 0$