The line mx-y-m-2=0 meets the circle x^2+y^2-4x+6y-3=0 at A and B.The lines joining A and B to the centre meet the circle again at C and D.If m is the parameter and set of lines CD are concurrent at p(a,b) then the value of a-b equals.

Asked by Sunil Soni | 23rd Sep, 2015, 10:29: AM

Expert Answer:

 
T h e space e q u a t i o n space o f space t h e space c i r c l e space i n space s tan d a r d space f o r m space i s open parentheses x minus 2 close parentheses squared plus open parentheses y plus 3 close parentheses squared equals 4 squared. T h e space c o o r d i n a t e s space o f space t h e space c e n t e r space a r e space left parenthesis 2 comma space minus 3 right parenthesis T h e space e n d s space o f space t h e space c h o r d space A B space a n d space t h e space c e n t e r space o f space t h e space c i r c l e space a r e space j o i n e d space t o space g e t space C space a n d space D space r e s p e c t i v e l y. space H e n c e comma space A c space a n d space B D space a r e space t h e space d i a m e t e r s. A l s o. space t h e space t r i a n g l e s space A O B space a n d space C O D space a r e space c o n g r u e n t. H e n c e comma space A B vertical line vertical line C D. E q u a t i o n space o f space A B space i s space y equals m x minus m minus 2 L e t space t h e space e q u a t i o n space o f space C D space b e space y equals m x plus K S i n c e comma space i t space p a s s e s space t h r o u g h space left parenthesis a comma space b right parenthesis comma space w e space g e t b equals m a plus K rightwards double arrow K equals b minus m a H e n c e comma space e q u a t i o n space o f space C D space i s space y equals m x plus b minus m a. H e r e space apostrophe m apostrophe space i s space a space p a r a m e t e r space w h i c h space c a n space t a k e space d i f f e r e n t space v a l u e s. F o r space m equals 1 comma space e q u a t i o n space o f space C D space i s space y equals x plus b minus a E q u a t i o n space o f space A B space i s space y equals x minus 3 D i s tan c e space o f space y equals x minus 3 space f r o m space left parenthesis 2 comma space minus 3 right parenthesis space i s space open vertical bar fraction numerator 2 minus open parentheses negative 3 close parentheses minus 3 over denominator square root of 1 squared plus 1 squared end root end fraction close vertical bar equals fraction numerator 2 over denominator square root of 2 end fraction D i s tan c e space o f space C D space f r o m space left parenthesis 2 comma space minus 3 right parenthesis space i s open vertical bar fraction numerator 2 minus open parentheses negative 3 close parentheses plus b minus a over denominator square root of 1 squared plus 1 squared end root end fraction close vertical bar equals fraction numerator 1 over denominator square root of 2 end fraction open vertical bar 5 plus b minus a close vertical bar B o t h space t h e space d i a tan c e s space a r e space e q u a l. H e n c e comma space fraction numerator 2 over denominator square root of 2 end fraction equals fraction numerator 1 over denominator square root of 2 end fraction open vertical bar 5 plus b minus a close vertical bar rightwards double arrow open vertical bar 5 plus b minus a close vertical bar equals 2 rightwards double arrow b minus a equals negative 5 plus-or-minus 2 rightwards double arrow b minus a equals negative 7 space left parenthesis r e j e c t i n g space b minus a equals negative 3 space sin c e space i t space g i v e s space t h e space e q u a t i o n space o f space A B right parenthesis H e n c e comma space a minus b equals 7

Answered by satyajit samal | 24th Sep, 2015, 12:04: PM