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Vector Algebra

Question
CBSEENMA12032924

Represent the following family of curves by forming the corresponding differential equations (a. b: parameters).
left parenthesis straight y minus straight b right parenthesis squared space equals space 4 left parenthesis straight x minus straight a right parenthesis






Solution

The equation of given curve is
                                   left parenthesis straight x minus straight b right parenthesis squared space space equals space 4 left parenthesis straight x minus straight a right parenthesis                           ...(1)
         Differentiating w.r.t.x, we get, 
                     2 left parenthesis straight y minus straight b right parenthesis dy over dx space equals space 4     or    left parenthesis straight y minus straight b right parenthesis space dy over dx equals 2       ....(2)
Again differentiating both sides w.r.t x we get,
         left parenthesis straight y minus straight b right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction plus dy over dx. dy over dx space space equals space 0 space space space space space or space space space fraction numerator 2 over denominator begin display style dy over dx end style end fraction fraction numerator straight d squared straight y over denominator dx squared end fraction plus open parentheses dy over dx close parentheses squared space equals space 0   open square brackets because space of space left parenthesis 2 right parenthesis close square brackets
or           2 fraction numerator straight d squared straight y over denominator dx squared end fraction plus open parentheses dy over dx close parentheses cubed space equals space 0 comma which is required differential equation.