obtain the  differential equation of the family of circles passing through the point (a,0) and (-a,0)

Asked by ppratim02 | 16th Nov, 2016, 05:01: PM

Expert Answer:

begin mathsize 18px style Since space the space circles space pass space through space the space points space left parenthesis straight a comma 0 right parenthesis space and space left parenthesis negative straight a comma 0 right parenthesis comma space the space centre space of space the space circle space will space be space the space origin space
and space the space circle space will space have space radius space equals space straight a
So comma space the space general space equation space of space such space straight a space circle space is space
open parentheses straight x minus straight a close parentheses squared plus straight y squared equals straight a squared space..... left parenthesis straight i right parenthesis
The space equation space should space be space differentiated space once space since space there space is space only space one space arbitrary space constant comma space straight a.
Differentiating space left parenthesis straight i right parenthesis space on space both space sides comma space we space get
2 left parenthesis straight x minus straight a right parenthesis plus 2 straight y dy over dx equals 0 space..... left parenthesis ii right parenthesis
rightwards double arrow straight x minus straight a equals negative straight y dy over dx space space and space space straight a equals straight x plus straight y dy over dx
Substituting space the space values space of space straight x minus straight a space and space straight a space in space left parenthesis straight i right parenthesis comma space we space get
open parentheses straight y dy over dx close parentheses squared plus straight y squared equals open parentheses straight x plus straight y dy over dx close parentheses squared space
rightwards double arrow straight y squared open parentheses dy over dx close parentheses squared plus straight y squared equals straight x squared plus 2 xy dy over dx plus straight y squared open parentheses dy over dx close parentheses squared
rightwards double arrow straight y squared equals straight x squared plus 2 xy dy over dx
which space is space the space required space differential space equation. end style

Answered by Rebecca Fernandes | 17th Nov, 2016, 09:08: AM