CBSE Class 12-commerce Answered

Obtain the differential equation of all circles of radius r

Asked by anjali | 03 Mar, 2017, 06:01: PM

Expert Answer

$begin mathsize 16px style Given space the space equation space of space family space of space circle comma space left parenthesis straight x minus straight a right parenthesis squared plus left parenthesis straight y minus straight b right parenthesis squared equals straight r squared space.... left parenthesis 1 right parenthesis Clearly comma space the space given space equation space have space two space arbirtrary space constants comma space so space we space differentiate space twice space wrt space straight x rightwards double arrow 2 left parenthesis straight x minus straight a right parenthesis plus 2 left parenthesis straight y minus straight b right parenthesis equals 0 rightwards double arrow left parenthesis straight x minus straight a right parenthesis plus left parenthesis straight y minus straight b right parenthesis dy over dx equals 0 space rightwards double arrow left parenthesis straight x minus straight a right parenthesis equals negative straight y minus straight b right parenthesis dy over dx.... left parenthesis 2 right parenthesis Differentiating space again space wrt space straight x 1 space space plus space left parenthesis straight y minus straight b right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction plus open parentheses dy over dx close parentheses squared equals 0 rightwards double arrow space left parenthesis straight y minus straight b right parenthesis equals negative fraction numerator 1 plus open parentheses dy over dx close parentheses squared over denominator fraction numerator straight d squared straight y over denominator dx squared end fraction end fraction space.... left parenthesis 3 right parenthesis Substitute space left parenthesis 3 right parenthesis space and space left parenthesis 2 right parenthesis space in space left parenthesis 1 right parenthesis comma space we space left parenthesis straight x minus straight a right parenthesis squared plus left parenthesis straight y minus straight b right parenthesis squared equals straight r squared rightwards double arrow open square brackets fraction numerator 1 plus open parentheses dy over dx close parentheses squared over denominator fraction numerator straight d squared straight y over denominator dx squared end fraction end fraction space dy over dx close square brackets squared space plus space fraction numerator open square brackets 1 plus open parentheses dy over dx close parentheses squared close square brackets squared over denominator fraction numerator straight d squared straight y over denominator dx squared end fraction end fraction equals straight r squared rightwards double arrow open square brackets 1 plus open parentheses dy over dx close parentheses squared close square brackets squared space open square brackets 1 plus open parentheses dy over dx close parentheses squared close square brackets equals straight r squared space open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses squared rightwards double arrow open square brackets 1 plus open parentheses dy over dx close parentheses squared close square brackets cubed equals straight r squared space open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses squared comma space which space is space the space required space DE. end style$