The equation of the circle having pair of lines x^2+2xy+3x+6y=0 as its normals and having the size just sufficient to contain the circle x(x-4)+y(y-3)=0is

Asked by Sunil Soni | 21st Sep, 2015, 04:45: PM

Expert Answer:

C o n s i d e r space t h e space p a i r space o f space l i n e s space x squared plus 2 x y plus 3 x plus 6 y equals 0 F a c t o r i z i n g space t h e space a b o v e space e q u a t i o n comma space w e space h a v e comma open parentheses x plus 2 y close parentheses open parentheses x plus 3 close parentheses equals 0 rightwards double arrow x plus 2 y equals 0 space o r space x plus 3 equals 0 rightwards double arrow x equals negative 2 y space o r space x equals negative 3 rightwards double arrow x equals negative 3 space a n d space y equals 3 over 2 N o r m a l s space o f space a space c i r c l e space i n t e r s e c t space a t space t h e space c e n t r e space o f space t h a t space c i r c l e. T h u s comma space t h e space i n t e r s e c t i o n space p o i n t space o f space n o r m a l s space i s space t h e space c e n t r e space o f space t h e space r e q u i r e d space c i r c l e. rightwards double arrow C e n t r e space o f space t h e space r e q u i r e d space c i r c l e space i s space C subscript 1 space open parentheses negative 3 comma 3 over 2 close parentheses L e t space R space b e space t h e space r a d i u s space o f space t h e space r e q u i r e d space c i r c l e.  C o n s i d e r space t h e space e q u a t i o n space o f space t h e space g i v e n space c i r c l e comma x open parentheses x minus 4 close parentheses plus y open parentheses y minus 3 close parentheses equals 0... left parenthesis 1 right parenthesis W h e n space open parentheses x subscript 1 comma y subscript 1 close parentheses space a n d space open parentheses x subscript 2 comma y subscript 2 close parentheses space a r e space t h e space e n d space p o i n t s space o f space t h e space d i a m t e r space o f space a space c i r c l e comma space t h e n space i t s space e q u a t i o n space i s open parentheses x minus x subscript 1 close parentheses open parentheses x minus x subscript 2 close parentheses plus open parentheses y minus y subscript 1 close parentheses open parentheses y minus y subscript 2 close parentheses equals 0 C o m p a r i n g space e q u a t i o n space left parenthesis 1 right parenthesis space w i t h space t h e space a b o v e space g e n e r a l space e q u a t i o n comma space w e space h a v e comma x subscript 1 equals 0 comma y subscript 1 equals 0 space a n d space x subscript 2 equals 4 space a n d space y subscript 2 equals 3 L e t space P open parentheses 0 comma 0 close parentheses space a n d space Q open parentheses 4 comma 3 close parentheses space b e space t h e space e n d space p o i n t s. T h u s comma space t h e space r a d i u s space o f space t h e space g i v e n space c i r c l e space i s space fraction numerator P Q over denominator 2 end fraction P Q equals square root of open parentheses 4 minus 0 close parentheses squared plus open parentheses 3 minus 0 close parentheses squared end root equals 5 T h u s comma space r a d i u s space o f space t h e space g i v e n space c i r c l e comma space r equals fraction numerator P Q over denominator 2 end fraction equals 5 over 2 C e n t r e space o f space t h e space g i v e n space c i r c l e space i s space fraction numerator P plus Q over denominator 2 end fraction T h u s comma space C e n t r e space o f space t h e space g i v e n space c i r c l e space i s space C subscript 2 space open parentheses fraction numerator 0 plus 4 over denominator 2 end fraction comma fraction numerator 0 plus 3 over denominator 2 end fraction close parentheses equals open parentheses 2 comma 3 over 2 close parentheses S i n c e space t h e space g i v e n space c i r c l e space i s space j u s t space c o n t a i n i n g space t h e space r e q u i r e d space c i r c l e comma space w e space h a v e comma C subscript 1 C subscript 2 equals R minus r C subscript 1 C subscript 2 equals square root of open parentheses 2 plus 3 close parentheses squared plus open parentheses 3 over 2 minus 3 over 2 close parentheses squared end root equals 5 R minus r equals R minus 5 over 2 A n d comma space sin c e space C subscript 1 C subscript 2 equals R minus r comma space w e space h a v e comma R minus 5 over 2 equals 5 rightwards double arrow R equals 5 plus 5 over 2 rightwards double arrow R equals 15 over 2 T h u s comma space t h e space e q u a t i o n space o f space t h e space r e q u i r e d space c i r c l e space i s space open parentheses x minus open parentheses negative 3 close parentheses close parentheses cubed plus open parentheses y minus 3 over 2 close parentheses squared equals open parentheses 15 over 2 close parentheses squared S i m p l i f y i n g space w e space h a v e x squared plus y squared plus 6 x minus 3 y minus 45 equals 0

Answered by Vimala Ramamurthy | 22nd Sep, 2015, 10:19: AM